Invariants for one form, for several forms.

Poincaré series for one binary form (this page), for two binary forms, for more binary forms.

# The Poincaré series

The Poincaré series of a graded module M = ∑ Mi is defined as P(t) = ∑ dim(Mi) ti.

Consider the graded algebra of the SL2(C)-invariant polynomials on the vector space of binary forms of degree n (graded by degree). There are several algorithms to compute its Poincaré series P(t) = Pn(t), all fast on a modern machine. Below a table with results. For the case of multiple binary forms, see the next page.

## Cayley-Sylvester

Let m(n,i) be the dimension of the vector space of invariant polynomial functions of degree i on the space of binary forms of degree n, so that P(t) = ∑ m(n,i) ti. Then m(n,i) = #(n,i,in/2) – #(n,i,in/2 – 1), where #(n,i,a) is the number of ways the number a can be written as a sum of i nonnegative integers, each at most n. (See Sylvester.)

Equivalently, m(n,i) = s(n+i,i,in/2) – s(n+i,i,in/2 – 1) where s(n,i,j) is the coefficient of qj in the q-binomial coefficient [n choose i]q (and is zero if j is not a nonnegative integer).

[Note that #(n,i,a) = s(n+i,i,a) since both satisfy the recursion #(n,i,a) = #(n–1,i,a) + #(n,i–1,a–n).]

Equivalently: m(n,i) is the coefficient of qin/2 in (1–q)[n+i choose i]q.

For example, for n=4 we see that [4 choose 0]q = 1, (1–q)[5 choose 1]q = 1–q5, (1–q)[6 choose 2]q = (1–q6)(1–q5)/(1–q2) = 1+q2+q4–q5–q7–q9, etc., so that P(t) = ∑ m(n,i) ti = 1 + 0t + t2 + ...

Since [n+i choose i]q = [n+i choose n]q, one has Hermite reciprocity: the coefficient of tm in Pn(t) is the same as the coefficient of tn in Pm(t).

## Springer

Cayley-Sylvester gives a power series form. A closed form was given by Springer and implemented by Brouwer & Cohen.

## Molien-Weyl

The Cayley-Sylvester formula can be formulated as: P(t) is the constant term in the Laurent series (in powers of q) of (1 – q2) ∏0≤j≤n (1 – q2j–nt)–1. This leads to the integral formulation: P(t) = (1 / 2πi) ∮ z–1 f(z,t) dz (integral around the unit circle, |t| < 1), where f(z,t) = (1 – z–2) ∏0≤j≤n (1 – zn–2jt)–1.

## Ðoković

Ðoković gives some conjectures and a fast algorithm depending on these conjectures. (The conjectures have now been proved, but it seems that the Springer algorithm is much faster.) His tables agree with those below.

More precisely, Ðoković puts for odd n = 2s–1: p(z,t) = ∏1≤i≤s (1 – z2i–1t) and q(z,t) = ∏1≤i≤s (z2i–1 – t) and r(t) = ∏2≤i≤n–1 (1 – t2i) and m = s2, and for even n = 2s: p(z,t) = ∏1≤i≤s (1 – z2it) and q(z,t) = ∏1≤i≤s (z2i – t) and r(t) = (1 + t) ∏2≤i≤n–1 (1 – ti) and m = s(s+1). (Now m is the z-degree of p(z,t) and q(z,t).) He conjectures for φ(z,t) = zm–2(z2–1)r(t) that there are polynomials a(z,t) and b(z,t) such that φ(z,t) = a(z,t)q(z,t) + b(z,t)p(z,t) (and this is indeed the case), and gives a simple expression for P(t) assuming this. (In the same paper there are further conjectures, incorrect for the case 4|n, n>4.)

## Properties

Let n ≥ 3, and write P(t) = A(t)/B(t) with coprime integral polynomials A(t) and B(t). Then the multiplicity of t = 1 as root of B(t) is n–2. The degree of B(t) is n+1 larger than the degree of A(t). The polynomial A(t) is palindromic. Indeed, P(1/t) = (–1)^(n–2) t^(n+1) P(t).

Dixmier shows that P(t) = A(t)/B(t) where A(t) is an integral polynomial, and B(t) equals 1–t2 if n=2, and ∏2≤i≤n–1 (1–t2i) if n is odd, and (1+t) ∏2≤i≤n–1 (1–ti) if n ≡ 2 (mod 4), n > 2, and (1+t) ∏2≤i≤n–1 (1–ti) / (1+t^(n/2–1)) if n ≡ 0 (mod 4), n > 0.

The polynomial A(t) = ∑ ai ti has a3 < 0 for n ≡ 2 (mod 4), n > 2 (as follows from Hermite reciprocity), and has otherwise a3 < 0 for n=8 only, a4 < 0 for n=6,8,10 only, a5 < 0 for n=10 only, a6 < 0 for n=5,7,9,11,13 only, a10 < 0 for n=7. For n ≤ 60 there are no negative coefficients other than these and the ones that follow since A(t) is a palindrome.

## Conjectures

Dixmier conjectures that the above A(t) and B(t) are coprime. (And this is true for n ≤ 60.) See also Derksen.

Ðoković adds the conjecture that in this coprime form the numerator is irreducible. (And also this is true for n ≤ 60.)

Dixmier also conjectures that if one writes P(t) = A*(t)/B*(t), where A*(t) is an integral polynomial, and B*(t) equals B(t) if n=2 or n is odd, is (1+t3) ∏2≤i≤n–1 (1–ti) if n ≡ 2 (mod 4), n > 2, and is ∏2≤i≤n–1 (1–ti) if n ≡ 0 (mod 4), n > 0, then A*(t) has nonnegative coefficients for n > 14. (Again, this holds for n ≤ 60.)

Furthermore, he conjectures that the algebra of invariants has a hsop with degrees 4,6,8,...,2n–2 if n is odd, n ≥ 15, with degrees 2,4,5,6,6,7,8,9,...,n–1 is n ≡ 2 (mod 4), n ≥ 18, and with degrees 2,3,4,...,n–1 is n ≡ 0 (mod 4). This implies the previous conjecture about positivity of the coefficients of A*(t).

### Note

Do not misinterpret the table below. The entries are not maximally simplified, but are chosen in the form P(t) = f(t)/(1–t^e)...(1–t^h) where the numerator has nonnegative coefficients only. If there is a homogeneous system of parameters of degrees e, ..., h then one expects such a form, but the converse is not necessarily true since there may be several ways of writing P(t) in this form.

For example, for n=12 one has
P(t)  = (1 + t^4 + t^5 + 3t^6 + 4t^7 + 7t^8 + 9t^9 + 17t^10 + 21t^11 + 36t^12 + 45t^13 + 65t^14 + 81t^15 + 110t^16 + 131t^17 + 168t^18 + 193t^19 + 232t^20 + 256t^21 + 293t^22 + 307t^23 + 336t^24 + 339t^25 + 351t^26 + 339t^27 + ... + t^52) / (1 – t^2)(1 – t^3)(1 – t^4)(1 – t^5)(1 – t^6)(1 – t^7)(1 – t^8)(1 – t^9)(1 – t^10)(1 – t^11) =
(1 + t^3 + t^4 + 3t^6 + 5t^7 + 7t^8 + 11t^9 + 21t^10 + 25t^11 + 40t^12 + 55t^13 + 75t^14 + 96t^15 + 130t^16 + 156t^17 + 194t^18 + 228t^19 + 267t^20 + 294t^21 + 330t^22 + 345t^23 + 364t^24 + 365t^25 + 364t^26 + ... + t^50) / (1 – t^2)(1 – t^4)(1 – t^5)^2(1 – t^6)^2(1 – t^7)(1 – t^8)(1 – t^9)(1 – t^11).

Now any hsop for n=12 contains an element of degree divisible by 10, so this latter fraction does not correspond to a hsop.

For n=7 there are two minimal ways of writing P(t):
P(t)  = (1 + 2t^8 + 4t^12 + 4t^14 + 5t^16 + 9t^18 + 6t^20 + 9t^22 + 8t^24 + 9t^26 + 6t^28 + 9t^30 + 5t^32 + 4t^34 + 4t^36 + 2t^40 + t^48) / (1 – t^4)(1 – t^8)(1 – t^12)^2(1 – t^20) =
(1 + t^8 + 5t^12 + 4t^14 + 3t^16 + 9t^18 + 4t^20 + 5t^22 + 8t^24 + 4t^26 + 4t^28 + 8t^30 + 5t^32 + 4t^34 + 9t^36 + 3t^38 + 4t^40 + 5t^42 + t^46 + t^54) / (1 – t^4)(1 – t^8)^2(1 – t^12)(1 – t^30)

and both correspond to a homogeneous system of parameters (Dixmier). For n=9 there are five minimal ways of writing P(t) and all five correspond to a homogeneous system of parameters (Brouwer & Popoviciu).

### Earlier computations

The cases n ≤ 10 and n = 12 were done by Sylvester & Franklin. The cases n ≤ 12 were done by K. Petr and J. Jarušek in a series of papers in the journal Rozprawy II, Třídy České. Akad. 56-57 (1946/1947). The cases n ≤ 16 were done by Brouwer & Cohen [loc.cit.] (in spite of the title). The cases n ≤ 30 were done by Ðoković [loc.cit.] assuming the truth of some conjectures, and the results for n ≤ 20 were given. (We did n ≤ 60 and give n ≤ 30.)

### General case

Bedratyuk has given formulas for ternary and for m-ary forms.

## References

A. Blokhuis, A. E. Brouwer, T. Szőnyi, Proof of a conjecture by Ðoković on the Poincaré series of the invariants of a binary form, preprint 2009.

A. E. Brouwer & A. M. Cohen, The Poincaré series of the polynomials invariant under SU2 in its irreducible representation of degree ≤17, report ZW134, Math. Centr. Amsterdam, Dec. 1979.

A. E. Brouwer & M. Popoviciu, The invariants of the binary nonic, J. Symb. Comput. 45 (2010) 709-720.

Harm Derksen, Universal denominators of Hilbert series, J. Algebra 285 (2005) 586-607.

J. Dixmier, Série de Poincaré et systèmes de paramètres pour les invariants des formes binaires de degré 7, Bull. Soc. Math. France 110 (1982) 303-318.

Jacques Dixmier, Quelques résultats et conjectures concernant les séries de Poincaré des invariants des formes binaires, pp. 127-160 in: Séminaire d'algèbre Paul Dubreil et Marie-Paule Malliavin (1983-1984), Springer LNM 1146, 1985.

D. Ž. Ðoković, A heuristic algorithm for computing the Poincaré series of the invariants of binary forms, Int. J. Contemp. Math. Sci. 1 (2006) 557-568. arXiv

F. Franklin, On the calculation of the generating functions and tables of grundforms for binary quantics, Amer. J. Math. 3 (1880) 128-153.

J. Jarušek, Vytvořující funkce pro počet invariantních útvarů binární formy 11. řádu v Petrově normálním tvaru, Rozpravy II. Třídy České Akad. 57 (1947), no. 8, 17 pp.

K. Petr, O vatvořující funkci v normálním tvaru pro počet invariantních útvarů u formy binární 12. stupně, Rozpravy II. Třídy České Akad. 56 (1946), no. 10, 16 pp.

T. A. Springer, On the invariant theory of SU2, Indag. Math. 42 (1980) 339-345.

J. J. Sylvester, Proof of the hitherto undemonstrated fundamental theorem of invariants, Philos. Mag. 5 (1878) 178-188.

J. J. Sylvester & F. Franklin, Tables of generating functions and grundforms for the binary quantics of the first ten orders, Amer. J. Math. 2 (1879) 223-251.

J. J. Sylvester & F. Franklin, Tables of the generating functions and groundforms of the binary duodecimic, with some general remarks, and tables of the irreducible syzygies of certain quantics, Amer. J. Math. 4 (1881) 41-61.

## Table

We consider three types of forms for P(t): the coprime form A(t)/B(t) with coprime polynomials A(t) and B(t) (now the Dixmier conjectures given above say what B(t) should be), the simplified form where the denominator is the above B*(t), and the representative forms, where the denominator is a product of factors (1 – t^e) and the numerator has nonnegative coefficients only. For n < 15 the behaviour is a bit erratic. Conjecturally, the simplified form is representative for all n ≥ 15. It certainly is for 15 ≤ n ≤ 60.

[The simplified form is the simplest way to "fix" the denominator B(t): If n≡2 (mod 4), n>2, there is no invariant of degree 3, and we have to remove the factors (1 + t) and (1 – t^3). The cheapest way is to replace them by a factor (1 – t^6), that is, by multiplying numerator and denominator by (1 – t + t^2). If n≡0 (mod 4), we have to remove the factor (1+t) and need a generator of degree divisible by n–2. The cheapest solution is to replace (1 + t)(1 – t^(n/2–1)) by (1 – t^(n–2)).]

### n=0

1 / (1 – t)

= 1 + t + t^2 + t^3 + t^4 + t^5 + ...

1

### n=2

1 / (1 – t^2)

= 1 + t^2 + t^4 + t^6 + t^8 + t^10 + t^12 + t^14 + t^16 + t^18 + t^20 + ...

For n ≥ 3, the rational function P(t) has degree –n–1.

### n=3

1 / (1 – t^4)

= 1 + t^4 + t^8 + t^12 + t^16 + t^20 + ...

### n=4

1 / (1 – t^2)(1 – t^3)

= 1 + t^2 + t^3 + t^4 + t^5 + 2t^6 + t^7 + 2t^8 + 2t^9 + 2t^10 + 2t^11 + 3t^12 + 2t^13 + 3t^14 + 3t^15 + 3t^16 + 3t^17 + 4t^18 + 3t^19 + 4t^20 + ...

### n=5

Simplified:
(1 – t^6 + t^12) / (1 – t^4)(1 – t^6)(1 – t^8)

Representative:
(1 + t^18) / (1 – t^4)(1 – t^8)(1 – t^12)

= 1 + t^4 + 2t^8 + 3t^12 + 4t^16 + t^18 + 5t^20 + ...

### n=6

Simplified:
(1 – t^5 + t^10) / (1 – t^2)(1 – t^4)(1 – t^5)(1 – t^6)

Representative:
(1 + t^15) / (1 – t^2)(1 – t^4)(1 – t^6)(1 – t^10)

= 1 + t^2 + 2t^4 + 3t^6 + 4t^8 + 6t^10 + 8t^12 + 10t^14 + t^15 + 13t^16 + t^17 + 16t^18 + 2t^19 + 20t^20 + ...

### n=7

Simplified:
(1 – t^6 + 2t^8 – t^10 + 5t^12 + 2t^14 + 6t^16 + 2t^18 + 5t^20 – t^22 + 2t^24 – t^26 + t^32) /
(1 – t^4)(1 – t^6)(1 – t^8)(1 – t^10)(1 – t^12)

Representative:
(1 + 2t^8 + 4t^12 + 4t^14 + 5t^16 + 9t^18 + 6t^20 + 9t^22 + 8t^24 + 9t^26 + 6t^28 + 9t^30 + 5t^32 + 4t^34 + 4t^36 + 2t^40 + t^48) /
(1 – t^4)(1 – t^8)(1 – t^12)^2(1 – t^20)

= 1 + t^4 + 4t^8 + 10t^12 + 4t^14 + 18t^16 + 13t^18 + 35t^20 + ...

### n=8

(1 + t^8 + t^9 + t^10 + t^18) / (1 – t^2)(1 – t^3)(1 – t^4)(1 – t^5)(1 – t^6)(1 – t^7)

= 1 + t^2 + t^3 + 2t^4 + 2t^5 + 4t^6 + 4t^7 + 7t^8 + 8t^9 + 12t^10 + 13t^11 + 20t^12 + 22t^13 + 31t^14 + 36t^15 + 47t^16 + 54t^17 + 71t^18 + 80t^19 + 102t^20 + ...

### n=9

Simplified:
(1 + t^4 – t^6 + 5t^8 + 3t^10 + 18t^12 + 15t^14 + 44t^16 + 43t^18 + 82t^20 + 76t^22 + 122t^24 + 107t^26 + 147t^28 + 119t^30 + 147t^32 + 107t^34 + 122t^36 + 76t^38 + 82t^40 + 43t^42 + 44t^44 + 15t^46 + 18t^48 + 3t^50 + 5t^52 – t^54 + t^56 + t^60) /
(1 – t^4)(1 – t^6)(1 – t^8)(1 – t^10)(1 – t^12)(1 – t^14)(1 – t^16)

Representative:
(1 + t^4 + 5t^8 + 4t^10 + 17t^12 + 20t^14 + 47t^16 + 61t^18 + 97t^20 + 120t^22 + 165t^24 + 189t^26 + 223t^28 + 241t^30 + 254t^32 + 254t^34 + 241t^36 + 223t^38 + 189t^40 + 165t^42 + 120t^44 + 97t^46 + 61t^48 + 47t^50 + 20t^52 + 17t^54 + 4t^56 + 5t^58 + t^62 + t^66) /
(1 – t^4)(1 – t^8)(1 – t^10)(1 – t^12)^2(1 – t^14)(1 – t^16)

= 1 + 2t^4 + 8t^8 + 5t^10 + 28t^12 + 27t^14 + 84t^16 + 99t^18 + 217t^20 + ...

### n=10

Simplified:
(1 – t^5 + 2t^6 – t^7 + 4t^8 + 4t^9 + 8t^10 + 6t^11 + 16t^12 + 9t^13 + 17t^14 + 15t^15 + 19t^16 + 12t^17 + 23t^18 + 12t^19 + 19t^20 + 15t^21 + 17t^22 + 9t^23 + 16t^24 + 6t^25 + 8t^26 + 4t^27 + 4t^28 – t^29 + 2t^30 – t^31 + t^36) /
(1 – t^2)(1 – t^4)(1 – t^5)(1 – t^6)^2(1 – t^7)(1 – t^8)(1 – t^9)

Representative:
(1 + 2t^6 + 4t^8 + 4t^9 + 7t^10 + 8t^11 + 15t^12 + 15t^13 + 20t^14 + 27t^15 + 29t^16 + 35t^17 + 40t^18 + 44t^19 + 47t^20 + 55t^21 + 52t^22 + 57t^23 + 56t^24 + 57t^25 + 52t^26 + 55t^27 + 47t^28 + 44t^29 + 40t^30 + 35t^31 + 29t^32 + 27t^33 + 20t^34 + 15t^35 + 15t^36 + 8t^37 + 7t^38 + 4t^39 + 4t^40 + 2t^42 + t^48) /
(1 – t^2)(1 – t^4)(1 – t^6)^2(1 – t^8)(1 – t^9)(1 – t^10)(1 – t^14)

= 1 + t^2 + 2t^4 + 6t^6 + 12t^8 + 5t^9 + 24t^10 + 13t^11 + 52t^12 + 33t^13 + 97t^14 + 80t^15 + 177t^16 + 160t^17 + 319t^18 + 301t^19 + 540t^20 + ...

### n=11

Simplified:
(1 + t^4 – t^6 + 10t^8 + 11t^10 + 58t^12 + 85t^14 + 222t^16 + 336t^18 + 660t^20 + 951t^22 + 1589t^24 + 2154t^26 + 3188t^28 + 4080t^30 + 5510t^32 + 6633t^34 + 8310t^36 + 9443t^38 + 11059t^40 + 11894t^42 + 13094t^44 + 13319t^46 + 13852t^48 + 13319t^50 + 13094t^52 + 11894t^54 + 11059t^56 + 9443t^58 + 8310t^60 + 6633t^62 + 5510t^64 + 4080t^66 + 3188t^68 + 2154t^70 + 1589t^72 + 951t^74 + 660t^76 + 336t^78 + 222t^80 + 85t^82 + 58t^84 + 11t^86 + 10t^88 – t^90 + t^92 + t^96) /
(1 – t^4)(1 – t^6)(1 – t^8)(1 – t^10)(1 – t^12)(1 – t^14)(1 – t^16)(1 – t^18)(1 – t^20)

Representative:
(1 + t^4 + 10t^8 + 12t^10 + 57t^12 + 95t^14 + 233t^16 + 394t^18 + 745t^20 + 1173t^22 + 1925t^24 + 2814t^26 + 4139t^28 + 5669t^30 + 7664t^32 + 9821t^34 + 12390t^36 + 14953t^38 + 17692t^40 + 20204t^42 + 22537t^44 + 24378t^46 + 25746t^48 + 26413t^50 + 26413t^52 + 25746t^54 + 24378t^56 + 22537t^58 + 20204t^60 + 17692t^62 + 14953t^64 + 12390t^66 + 9821t^68 + 7664t^70 + 5669t^72 + 4139t^74 + 2814t^76 + 1925t^78 + 1173t^80 + 745t^82 + 394t^84 + 233t^86 + 95t^88 + 57t^90 + 12t^92 + 10t^94 + t^98 + t^102) /
(1 – t^4)(1 – t^8)(1 – t^10)(1 – t^12)^2(1 – t^14)(1 – t^16)(1 – t^18)(1 – t^20)

= 1 + 2t^4 + 13t^8 + 13t^10 + 73t^12 + 110t^14 + 320t^16 + 529t^18 + 1160t^20 + ...

### n=12

(1 + t^4 + t^5 + 3t^6 + 4t^7 + 7t^8 + 9t^9 + 17t^10 + 21t^11 + 36t^12 + 45t^13 + 65t^14 + 81t^15 + 110t^16 + 131t^17 + 168t^18 + 193t^19 + 232t^20 + 256t^21 + 293t^22 + 307t^23 + 336t^24 + 339t^25 + 351t^26 + 339t^27 + 336t^28 + 307t^29 + 293t^30 + 256t^31 + 232t^32 + 193t^33 + 168t^34 + 131t^35 + 110t^36 + 81t^37 + 65t^38 + 45t^39 + 36t^40 + 21t^41 + 17t^42 + 9t^43 + 7t^44 + 4t^45 + 3t^46 + t^47 + t^48 + t^52) /
(1 – t^2)(1 – t^3)(1 – t^4)(1 – t^5)(1 – t^6)(1 – t^7)(1 – t^8)(1 – t^9)(1 – t^10)(1 – t^11)

= 1 + t^2 + t^3 + 3t^4 + 3t^5 + 8t^6 + 10t^7 + 20t^8 + 28t^9 + 52t^10 + 73t^11 + 127t^12 + 181t^13 + 291t^14 + 418t^15 + 639t^16 + 902t^17 + 1330t^18 + 1848t^19 + 2634t^20 + ...

### n=13

Simplified:
(1 + t^4 – t^6 + 19t^8 + 31t^10 + 157t^12 + 321t^14 + 885t^16 + 1756t^18 + 3794t^20 + 6856t^22 + 12788t^24 + 21324t^26 + 35633t^28 + 55326t^30 + 85174t^32 + 124064t^34 + 178645t^36 + 246238t^38 + 334814t^40 + 439321t^42 + 568305t^44 + 712862t^46 + 881834t^48 + 1061455t^50 + 1259989t^52 + 1459221t^54 + 1666984t^56 + 1860904t^58 + 2049854t^60 + 2209072t^62 + 2349306t^64 + 2446352t^66 + 2514111t^68 + 2530530t^70 + 2514111t^72 + 2446352t^74 + 2349306t^76 + 2209072t^78 + 2049854t^80 + 1860904t^82 + 1666984t^84 + 1459221t^86 + 1259989t^88 + 1061455t^90 + 881834t^92 + 712862t^94 + 568305t^96 + 439321t^98 + 334814t^100 + 246238t^102 + 178645t^104 + 124064t^106 + 85174t^108 + 55326t^110 + 35633t^112 + 21324t^114 + 12788t^116 + 6856t^118 + 3794t^120 + 1756t^122 + 885t^124 + 321t^126 + 157t^128 + 31t^130 + 19t^132 – t^134 + t^136 + t^140)/
(1 – t^4)(1 – t^6)(1 – t^8)(1 – t^10)(1 – t^12)(1 – t^14)(1 – t^16)(1 – t^18)(1 – t^20)(1 – t^22)(1 – t^24)

Representative:
(1 + t^4 + 19t^8 + 32t^10 + 156t^12 + 340t^14 + 916t^16 + 1913t^18 + 4115t^20 + 7741t^22 + 14544t^24 + 25118t^26 + 42489t^28 + 68114t^30 + 106498t^32 + 159697t^34 + 233971t^36 + 331412t^38 + 458878t^40 + 617966t^42 + 814543t^44 + 1047676t^46 + 1321155t^48 + 1629760t^50 + 1972851t^52 + 2341055t^54 + 2728439t^56 + 3120893t^58 + 3509075t^60 + 3876056t^62 + 4210210t^64 + 4496206t^66 + 4723183t^68 + 4879836t^70 + 4960463t^72 + 4960463t^74 + 4879836t^76 + 4723183t^78 + 4496206t^80 + 4210210t^82 + 3876056t^84 + 3509075t^86 + 3120893t^88 + 2728439t^90 + 2341055t^92 + 1972851t^94 + 1629760t^96 + 1321155t^98 + 1047676t^100 + 814543t^102 + 617966t^104 + 458878t^106 + 331412t^108 + 233971t^110 + 159697t^112 + 106498t^114 + 68114t^116 + 42489t^118 + 25118t^120 + 14544t^122 + 7741t^124 + 4115t^126 + 1913t^128 + 916t^130 + 340t^132 + 156t^134 + 32t^136 + 19t^138 + t^142 + t^146) /
(1 – t^4)(1 – t^8)(1 – t^10)(1 – t^12)^2 (1 – t^14)(1 – t^16)(1 – t^18)(1 – t^20)(1 – t^22)(1 – t^24)

= 1 + 2t^4 + 22t^8 + 33t^10 + 181t^12 + 375t^14 + 1120t^16 + 2342t^18 + 5467t^20 + ...

### n=14

Simplified:
(1 + t^4 – t^5 + 5t^6 + 3t^7 + 18t^8 + 21t^9 + 56t^10 + 72t^11 + 155t^12 + 209t^13 + 375t^14 + 523t^15 + 836t^16 + 1131t^17 + 1695t^18 + 2234t^19 + 3132t^20 + 4029t^21 + 5371t^22 + 6691t^23 + 8566t^24 + 10348t^25 + 12736t^26 + 14971t^27 + 17789t^28 + 20306t^29 + 23400t^30 + 25973t^31 + 29023t^32 + 31385t^33 + 34068t^34 + 35858t^35 + 37893t^36 + 38831t^37 + 39932t^38 + 39890t^39 + 39932t^40 + 38831t^41 + 37893t^42 + 35858t^43 + 34068t^44 + 31385t^45 + 29023t^46 + 25973t^47 + 23400t^48 + 20306t^49 + 17789t^50 + 14971t^51 + 12736t^52 + 10348t^53 + 8566t^54 + 6691t^55 + 5371t^56 + 4029t^57 + 3132t^58 + 2234t^59 + 1695t^60 + 1131t^61 + 836t^62 + 523t^63 + 375t^64 + 209t^65 + 155t^66 + 72t^67 + 56t^68 + 21t^69 + 18t^70 + 3t^71 + 5t^72 – t^73 + t^74 + t^78) /
(1 – t^2)(1 – t^4)(1 – t^5)(1 – t^6)^2(1 – t^7)(1 – t^8)(1 – t^9)(1 – t^10)(1 – t^11)(1 – t^12)(1 – t^13)

Representative:
(1 + t^4 + 5t^6 + 3t^7 + 18t^8 + 22t^9 + 55t^10 + 77t^11 + 158t^12 + 227t^13 + 396t^14 + 579t^15 + 908t^16 + 1286t^17 + 1904t^18 + 2609t^19 + 3655t^20 + 4865t^21 + 6502t^22 + 8386t^23 + 10800t^24 + 13480t^25 + 16765t^26 + 20342t^27 + 24480t^28 + 28872t^29 + 33748t^30 + 38709t^31 + 43994t^32 + 49174t^33 + 54374t^34 + 59258t^35 + 63866t^36 + 67854t^37 + 71317t^38 + 73958t^39 + 75790t^40 + 76724t^41 + 76724t^42 + 75790t^43 + 73958t^44 + 71317t^45 + 67854t^46 + 63866t^47 + 59258t^48 + 54374t^49 + 49174t^50 + 43994t^51 + 38709t^52 + 33748t^53 + 28872t^54 + 24480t^55 + 20342t^56 + 16765t^57 + 13480t^58 + 10800t^59 + 8386t^60 + 6502t^61 + 4865t^62 + 3655t^63 + 2609t^64 + 1904t^65 + 1286t^66 + 908t^67 + 579t^68 + 396t^69 + 227t^70 + 158t^71 + 77t^72 + 55t^73 + 22t^74 + 18t^75 + 3t^76 + 5t^77 + t^79 + t^83) /
(1 – t^2)(1 – t^4)(1 – t^6)^2(1 – t^7)(1 – t^8)(1 – t^9)(1 – t^10)^2 (1 – t^11)(1 – t^12)(1 – t^13)

= 1 + t^2 + 3t^4 + 10t^6 + 4t^7 + 31t^8 + 27t^9 + 97t^10 + 110t^11 + 291t^12 + 375t^13 + 802t^14 + 1111t^15 + 2077t^16 + 2930t^17 + 5034t^18 + 7120t^19 + 11463t^20 + ...

From here on, the pattern in the table is more regular.

If n is odd, the numerator of the representative form has degree (n–3)(n+1) and the denominator is (1 – t^4)(1 – t^6)(1 – t^8) ... (1 – t^(2n–2)), and these are coprime.

If n ≡ 2 (mod 4), the fraction in lowest terms has denominator (1 + t)(1 – t^2)(1 – t^3)(1 – t^4) ... (1 – t^(n–1)) and the cheapest way to turn that into an expression with nonnegative numerator and with denominator that is a product of factors (1 – t^e) is to multiply numerator and denominator with (1 – t + t^2), turning (1 + t)(1 – t^3) into (1 – t^6). The resulting numerator has degree (n–1)(n–2)/2.

If n ≡ 0 (mod 4), the fraction in lowest terms has denominator (1 + t).(1 – t^2)(1 – t^3)(1 – t^4) ... (1 – t^(n–4))(1 – t^(n–3)).(1 – t^((n–2)/2)).(1 – t^(n–1)) and the nicest way to turn that into an expression with nonnegative numerator and with denominator that is a product of factors (1 – t^e) is to multiply numerator and denominator with (1 – t + t^2 – ... + t^(n/2–2)) = (1 + t^(n/2–1))/(t+1) so that the denominator becomes (1 – t^2)(1 – t^3)(1 – t^4) ... (1 – t^(n–1)). The resulting numerator has degree (n+1)(n–4)/2.

### n=15

(1 + 2t^4 + 32t^8 + 76t^10 + 378t^12 + 995t^14 + 3048t^16 + 7294t^18 + 17681t^20 + 37736t^22 + 78903t^24 + 152321t^26 + 285968t^28 + 507762t^30 + 876759t^32 + 1451423t^34 + 2341739t^36 + 3653241t^38 + 5568497t^40 + 8254649t^42 + 11983447t^44 + 16987847t^46 + 23631274t^48 + 32196429t^50 + 43116834t^52 + 56681420t^54 + 73342055t^56 + 93320393t^58 + 117007543t^60 + 144461993t^62 + 175919353t^64 + 211175615t^66 + 250222591t^68 + 292516508t^70 + 337751801t^72 + 385016863t^74 + 433713649t^76 + 482605505t^78 + 530877973t^80 + 577086324t^82 + 620343376t^84 + 659172312t^86 + 692798202t^88 + 719914717t^90 + 740045690t^92 + 752239053t^94 + 756462172t^96 + 752239053t^98 + 740045690t^100 + 719914717t^102 + 692798202t^104 + 659172312t^106 + 620343376t^108 + 577086324t^110 + 530877973t^112 + 482605505t^114 + 433713649t^116 + 385016863t^118 + 337751801t^120 + 292516508t^122 + 250222591t^124 + 211175615t^126 + 175919353t^128 + 144461993t^130 + 117007543t^132 + 93320393t^134 + 73342055t^136 + 56681420t^138 + 43116834t^140 + 32196429t^142 + 23631274t^144 + 16987847t^146 + 11983447t^148 + 8254649t^150 + 5568497t^152 + 3653241t^154 + 2341739t^156 + 1451423t^158 + 876759t^160 + 507762t^162 + 285968t^164 + 152321t^166 + 78903t^168 + 37736t^170 + 17681t^172 + 7294t^174 + 3048t^176 + 995t^178 + 378t^180 + 76t^182 + 32t^184 + 2t^188 + t^192) / (1 – t^4)(1 – t^6)(1 – t^8)(1 – t^10)(1 – t^12)(1 – t^14)(1 – t^16) (1 – t^18)(1 – t^20)(1 – t^22)(1 – t^24)(1 – t^26)(1 – t^28)

= 1 + 3t^4 + t^6 + 36t^8 + 80t^10 + 418t^12 + 1111t^14 + 3581t^16 + 8899t^18 + 22786t^20 + ...

### n=16

(1 + t^4 + 2t^5 + 8t^6 + 11t^7 + 28t^8 + 51t^9 + 102t^10 + 177t^11 + 340t^12 + 561t^13 + 980t^14 + 1586t^15 + 2565t^16 + 3955t^17 + 6095t^18 + 8991t^19 + 13206t^20 + 18815t^21 + 26498t^22 + 36437t^23 + 49596t^24 + 66028t^25 + 87003t^26 + 112578t^27 + 144034t^28 + 181363t^29 + 226014t^30 + 277437t^31 + 337179t^32 + 404317t^33 + 479951t^34 + 562691t^35 + 653453t^36 + 749807t^37 + 852481t^38 + 958443t^39 + 1067723t^40 + 1176799t^41 + 1285637t^42 + 1389850t^43 + 1489589t^44 + 1580460t^45 + 1662409t^46 + 1731403t^47 + 1788102t^48 + 1828489t^49 + 1854175t^50 + 1862064t^51 + 1854175t^52 + 1828489t^53 + 1788102t^54 + 1731403t^55 + 1662409t^56 + 1580460t^57 + 1489589t^58 + 1389850t^59 + 1285637t^60 + 1176799t^61 + 1067723t^62 + 958443t^63 + 852481t^64 + 749807t^65 + 653453t^66 + 562691t^67 + 479951t^68 + 404317t^69 + 337179t^70 + 277437t^71 + 226014t^72 + 181363t^73 + 144034t^74 + 112578t^75 + 87003t^76 + 66028t^77 + 49596t^78 + 36437t^79 + 26498t^80 + 18815t^81 + 13206t^82 + 8991t^83 + 6095t^84 + 3955t^85 + 2565t^86 + 1586t^87 + 980t^88 + 561t^89 + 340t^90 + 177t^91 + 102t^92 + 51t^93 + 28t^94 + 11t^95 + 8t^96 + 2t^97 + t^98 + t^102) / (1 – t^2)(1 – t^3)(1 – t^4)(1 – t^5)(1 – t^6)(1 – t^7) (1 – t^8)(1 – t^9)(1 – t^10)(1 – t^11)(1 – t^12)(1 – t^13)(1 – t^14) (1 – t^15)

= 1 + t^2 + t^3 + 3t^4 + 4t^5 + 13t^6 + 18t^7 + 47t^8 + 84t^9 + 177t^10 + 320t^11 + 639t^12 + 1120t^13 + 2077t^14 + 3581t^15 + 6235t^16 + 10395t^17 + 17344t^18 + 27940t^19 + 44848t^20 + ...

### n=17

(1 + 2t^4 + 50t^8 + 156t^10 + 844t^12 + 2716t^14 + 9280t^16 + 26055t^18 + 70846t^20 + 173224t^22 + 405183t^24 + 883551t^26 + 1847356t^28 + 3669433t^30 + 7024773t^32 + 12919848t^34 + 23019526t^36 + 39697193t^38 + 66608244t^40 + 108748704t^42 + 173371011t^44 + 270001994t^46 + 411791616t^48 + 615371715t^50 + 902700319t^52 + 1300556398t^54 + 1842885348t^56 + 2569619659t^58 + 3529481145t^60 + 4777707107t^62 + 6379225544t^64 + 8404807944t^66 + 10934524315t^68 + 14051849433t^70 + 17847385164t^72 + 22410522390t^74 + 27833575972t^76 + 34200807232t^78 + 41593316021t^80 + 50075498973t^82 + 59701312756t^84 + 70498510402t^86 + 82477120228t^88 + 95612069584t^90 + 109854817368t^92 + 125115124389t^94 + 141276752124t^96 + 158179073390t^98 + 175637520454t^100 + 193425539961t^102 + 211299997259t^104 + 228983624510t^106 + 246195050087t^108 + 262631400199t^110 + 278002047047t^112 + 292010632283t^114 + 304391087240t^116 + 314888097678t^118 + 323293686822t^120 + 329425050810t^122 + 333160013567t^124 + 334411422423t^126 + 333160013567t^128 + 329425050810t^130 + 323293686822t^132 + 314888097678t^134 + 304391087240t^136 + 292010632283t^138 + 278002047047t^140 + 262631400199t^142 + 246195050087t^144 + 228983624510t^146 + 211299997259t^148 + 193425539961t^150 + 175637520454t^152 + 158179073390t^154 + 141276752124t^156 + 125115124389t^158 + 109854817368t^160 + 95612069584t^162 + 82477120228t^164 + 70498510402t^166 + 59701312756t^168 + 50075498973t^170 + 41593316021t^172 + 34200807232t^174 + 27833575972t^176 + 22410522390t^178 + 17847385164t^180 + 14051849433t^182 + 10934524315t^184 + 8404807944t^186 + 6379225544t^188 + 4777707107t^190 + 3529481145t^192 + 2569619659t^194 + 1842885348t^196 + 1300556398t^198 + 902700319t^200 + 615371715t^202 + 411791616t^204 + 270001994t^206 + 173371011t^208 + 108748704t^210 + 66608244t^212 + 39697193t^214 + 23019526t^216 + 12919848t^218 + 7024773t^220 + 3669433t^222 + 1847356t^224 + 883551t^226 + 405183t^228 + 173224t^230 + 70846t^232 + 26055t^234 + 9280t^236 + 2716t^238 + 844t^240 + 156t^242 + 50t^244 + 2t^248 + t^252) / (1 – t^4)(1 – t^6)(1 – t^8)(1 – t^10)(1 – t^12)(1 – t^14)(1 – t^16) (1 – t^18)(1 – t^20)(1 – t^22)(1 – t^24)(1 – t^26)(1 – t^28) (1 – t^30)(1 – t^32)

### n=18

(1 + 2t^4 + 10t^6 + 11t^7 + 51t^8 + 82t^9 + 230t^10 + 403t^11 + 918t^12 + 1638t^13 + 3242t^14 + 5649t^15 + 10254t^16 + 17174t^17 + 29206t^18 + 47052t^19 + 75877t^20 + 117780t^21 + 181801t^22 + 272539t^23 + 405174t^24 + 588402t^25 + 846221t^26 + 1193672t^27 + 1666785t^28 + 2289024t^29 + 3112096t^30 + 4169874t^31 + 5532493t^32 + 7245610t^33 + 9399502t^34 + 12050747t^35 + 15309825t^36 + 19240907t^37 + 23971143t^38 + 29566304t^39 + 36163387t^40 + 43819319t^41 + 52671150t^42 + 62753658t^43 + 74190106t^44 + 86978179t^45 + 101212342t^46 + 116836663t^47 + 133902919t^48 + 152287338t^49 + 171986679t^50 + 192800688t^51 + 214664688t^52 + 237298643t^53 + 260578657t^54 + 284151128t^55 + 307842228t^56 + 331241256t^57 + 354143274t^58 + 376103843t^59 + 396913842t^60 + 416125463t^61 + 433554690t^62 + 448784632t^63 + 461686799t^64 + 471905773t^65 + 479396248t^66 + 483890064t^67 + 485442519t^68 + 483890064t^69 + 479396248t^70 + 471905773t^71 + 461686799t^72 + 448784632t^73 + 433554690t^74 + 416125463t^75 + 396913842t^76 + 376103843t^77 + 354143274t^78 + 331241256t^79 + 307842228t^80 + 284151128t^81 + 260578657t^82 + 237298643t^83 + 214664688t^84 + 192800688t^85 + 171986679t^86 + 152287338t^87 + 133902919t^88 + 116836663t^89 + 101212342t^90 + 86978179t^91 + 74190106t^92 + 62753658t^93 + 52671150t^94 + 43819319t^95 + 36163387t^96 + 29566304t^97 + 23971143t^98 + 19240907t^99 + 15309825t^100 + 12050747t^101 + 9399502t^102 + 7245610t^103 + 5532493t^104 + 4169874t^105 + 3112096t^106 + 2289024t^107 + 1666785t^108 + 1193672t^109 + 846221t^110 + 588402t^111 + 405174t^112 + 272539t^113 + 181801t^114 + 117780t^115 + 75877t^116 + 47052t^117 + 29206t^118 + 17174t^119 + 10254t^120 + 5649t^121 + 3242t^122 + 1638t^123 + 918t^124 + 403t^125 + 230t^126 + 82t^127 + 51t^128 + 11t^129 + 10t^130 + 2t^132 + t^136) / (1 – t^2)(1 – t^4)(1 – t^5)(1 – t^6)^2 (1 – t^7)(1 – t^8)(1 – t^9)(1 – t^10)(1 – t^11)(1 – t^12)(1 – t^13) (1 – t^14)(1 – t^15)(1 – t^16)(1 – t^17)

### n=19

(1 + 2t^4 + t^6 + 76t^8 + 296t^10 + 1763t^12 + 6738t^14 + 25712t^16 + 82893t^18 + 252012t^20 + 694765t^22 + 1807368t^24 + 4392969t^26 + 10154779t^28 + 22296093t^30 + 46930799t^32 + 94802787t^34 + 184822672t^36 + 348299741t^38 + 636837951t^40 + 1131559371t^42 + 1959027689t^44 + 3309329549t^46 + 5465457626t^48 + 8835569146t^50 + 14002953513t^52 + 21778902561t^54 + 33281420196t^56 + 50015102190t^58 + 73986433683t^60 + 107815040622t^62 + 154890725357t^64 + 219515780752t^66 + 307103168571t^68 + 424341546490t^70 + 579425240340t^72 + 782222455286t^74 + 1044516156680t^76 + 1380143077777t^78 + 1805211986577t^80 + 2338168202449t^82 + 2999945060393t^84 + 3813898895206t^86 + 4805843429850t^88 + 6003797685274t^90 + 7437855144286t^92 + 9139694108669t^94 + 11142247265979t^96 + 13478942963820t^98 + 16183166152183t^100 + 19287230767266t^102 + 22821658964018t^104 + 26813925195712t^106 + 31287624164380t^108 + 36261071290807t^110 + 41746472475910t^112 + 47748507080547t^114 + 54263646677747t^116 + 61278874674037t^118 + 68771319597259t^120 + 76707272862777t^122 + 85042295486194t^124 + 93720678352722t^126 + 102676146969267t^128 + 111831862933442t^130 + 121101792448393t^132 + 130391278393710t^134 + 139599058259333t^136 + 148618354053166t^138 + 157339436459345t^140 + 165651094224795t^142 + 173443559767368t^144 + 180610146679903t^146 + 187050288671933t^148 + 192671089114197t^150 + 197390193869982t^152 + 201136986494180t^154 + 203855025614692t^156 + 205502649265697t^158 + 206054755643582t^160 + 205502649265697t^162 + 203855025614692t^164 + 201136986494180t^166 + 197390193869982t^168 + 192671089114197t^170 + 187050288671933t^172 + 180610146679903t^174 + 173443559767368t^176 + 165651094224795t^178 + 157339436459345t^180 + 148618354053166t^182 + 139599058259333t^184 + 130391278393710t^186 + 121101792448393t^188 + 111831862933442t^190 + 102676146969267t^192 + 93720678352722t^194 + 85042295486194t^196 + 76707272862777t^198 + 68771319597259t^200 + 61278874674037t^202 + 54263646677747t^204 + 47748507080547t^206 + 41746472475910t^208 + 36261071290807t^210 + 31287624164380t^212 + 26813925195712t^214 + 22821658964018t^216 + 19287230767266t^218 + 16183166152183t^220 + 13478942963820t^222 + 11142247265979t^224 + 9139694108669t^226 + 7437855144286t^228 + 6003797685274t^230 + 4805843429850t^232 + 3813898895206t^234 + 2999945060393t^236 + 2338168202449t^238 + 1805211986577t^240 + 1380143077777t^242 + 1044516156680t^244 + 782222455286t^246 + 579425240340t^248 + 424341546490t^250 + 307103168571t^252 + 219515780752t^254 + 154890725357t^256 + 107815040622t^258 + 73986433683t^260 + 50015102190t^262 + 33281420196t^264 + 21778902561t^266 + 14002953513t^268 + 8835569146t^270 + 5465457626t^272 + 3309329549t^274 + 1959027689t^276 + 1131559371t^278 + 636837951t^280 + 348299741t^282 + 184822672t^284 + 94802787t^286 + 46930799t^288 + 22296093t^290 + 10154779t^292 + 4392969t^294 + 1807368t^296 + 694765t^298 + 252012t^300 + 82893t^302 + 25712t^304 + 6738t^306 + 1763t^308 + 296t^310 + 76t^312 + t^314 + 2t^316 + t^320) / (1 – t^4)(1 – t^6)(1 – t^8)(1 – t^10)(1 – t^12) (1 – t^14)(1 – t^16)(1 – t^18)(1 – t^20)(1 – t^22)(1 – t^24) (1 – t^26)(1 – t^28)(1 – t^30)(1 – t^32)(1 – t^34)(1 – t^36)

### n=20

(1 + 2t^4 + 3t^5 + 14t^6 + 26t^7 + 74t^8 + 159t^9 + 386t^10 + 813t^11 + 1786t^12 + 3581t^13 + 7194t^14 + 13690t^15 + 25662t^16 + 46264t^17 + 81972t^18 + 140858t^19 + 237716t^20 + 391489t^21 + 633566t^22 + 1004435t^23 + 1567003t^24 + 2401414t^25 + 3626076t^26 + 5390337t^27 + 7904749t^28 + 11431403t^29 + 16326733t^30 + 23026390t^31 + 32104634t^32 + 44251748t^33 + 60350746t^34 + 81444897t^35 + 108834679t^36 + 144027146t^37 + 188856601t^38 + 245409166t^39 + 316164054t^40 + 403886629t^41 + 511790842t^42 + 643385302t^43 + 802659024t^44 + 993869808t^45 + 1221746711t^46 + 1491212905t^47 + 1807606172t^48 + 2176318945t^49 + 2603044019t^50 + 3093325449t^51 + 3652826842t^52 + 4286795701t^53 + 5000365547t^54 + 5797926021t^55 + 6683480280t^56 + 7659930019t^57 + 8729496251t^58 + 9892940351t^59 + 11150071817t^60 + 12498910419t^61 + 13936313872t^62 + 15457101887t^63 + 17054812157t^64 + 18720815924t^65 + 20445221332t^66 + 22215981071t^67 + 24019965060t^68 + 25842070888t^69 + 27666450024t^70 + 29475635493t^71 + 31251911254t^72 + 32976432897t^73 + 34630731555t^74 + 36195812801t^75 + 37653744237t^76 + 38986726195t^77 + 40178723548t^78 + 41214465019t^79 + 42081091938t^80 + 42767062988t^81 + 43263759348t^82 + 43564293507t^83 + 43665034627t^84 + 43564293507t^85 + 43263759348t^86 + 42767062988t^87 + 42081091938t^88 + 41214465019t^89 + 40178723548t^90 + 38986726195t^91 + 37653744237t^92 + 36195812801t^93 + 34630731555t^94 + 32976432897t^95 + 31251911254t^96 + 29475635493t^97 + 27666450024t^98 + 25842070888t^99 + 24019965060t^100 + 22215981071t^101 + 20445221332t^102 + 18720815924t^103 + 17054812157t^104 + 15457101887t^105 + 13936313872t^106 + 12498910419t^107 + 11150071817t^108 + 9892940351t^109 + 8729496251t^110 + 7659930019t^111 + 6683480280t^112 + 5797926021t^113 + 5000365547t^114 + 4286795701t^115 + 3652826842t^116 + 3093325449t^117 + 2603044019t^118 + 2176318945t^119 + 1807606172t^120 + 1491212905t^121 + 1221746711t^122 + 993869808t^123 + 802659024t^124 + 643385302t^125 + 511790842t^126 + 403886629t^127 + 316164054t^128 + 245409166t^129 + 188856601t^130 + 144027146t^131 + 108834679t^132 + 81444897t^133 + 60350746t^134 + 44251748t^135 + 32104634t^136 + 23026390t^137 + 16326733t^138 + 11431403t^139 + 7904749t^140 + 5390337t^141 + 3626076t^142 + 2401414t^143 + 1567003t^144 + 1004435t^145 + 633566t^146 + 391489t^147 + 237716t^148 + 140858t^149 + 81972t^150 + 46264t^151 + 25662t^152 + 13690t^153 + 7194t^154 + 3581t^155 + 1786t^156 + 813t^157 + 386t^158 + 159t^159 + 74t^160 + 26t^161 + 14t^162 + 3t^163 + 2t^164 + t^168) / (1 – t^2)(1 – t^3)(1 – t^4)(1 – t^5) (1 – t^6)(1 – t^7)(1 – t^8)(1 – t^9)(1 – t^10)(1 – t^11)(1 – t^12) (1 – t^13)(1 – t^14)(1 – t^15)(1 – t^16)(1 – t^17)(1 – t^18) (1 – t^19)

= 1 + t^2 + t^3 + 4t^4 + 5t^5 + 20t^6 + 35t^7 + 102t^8 + 217t^9 + 540t^10 + 1160t^11 + 2634t^12 + 5467t^13 + 11463t^14 + 22786t^15 + 44848t^16 + 85068t^17 + 159018t^18 + 288914t^19 + 516643t^20 + ...

### n=21

(1 + 3t^4 + 2t^6 + 112t^8 + 540t^10 + 3479t^12 + 15467t^14 + 65914t^16 + 240419t^18 + 813348t^20 + 2501947t^22 + 7196202t^24 + 19325255t^26 + 49083610t^28 + 118193567t^30 + 271792913t^32 + 598609706t^34 + 1268624162t^36 + 2593901734t^38 + 5133646455t^40 + 9856315443t^42 + 18402427436t^44 + 33473629379t^46 + 59430973423t^48 + 103147796363t^50 + 175263298262t^52 + 291908993401t^54 + 477148117122t^56 + 766230180514t^58 + 1210030918912t^60 + 1880803164406t^62 + 2879776991399t^64 + 4346732639516t^66 + 6472286726974t^68 + 9512958483185t^70 + 13809982226062t^72 + 19811780140866t^74 + 28101294636548t^76 + 39427794406359t^78 + 54744574508758t^80 + 75251660645652t^82 + 102445130580609t^84 + 138171399515663t^86 + 184688242166705t^88 + 244729831870232t^90 + 321577713371961t^92 + 419133599770660t^94 + 541996086847665t^96 + 695535474333511t^98 + 885969085020306t^100 + 1120429310910923t^102 + 1407027324715603t^104 + 1754902558254528t^106 + 2174261846638307t^108 + 2676396167205380t^110 + 3273680430050588t^112 + 3979542167320963t^114 + 4808406912033183t^116 + 5775604247640348t^118 + 6897245571573098t^120 + 8190055985475190t^122 + 9671175601669097t^124 + 11357911416698039t^126 + 13267460256461451t^128 + 15416582904745600t^130 + 17821255959669095t^132 + 20496280522892201t^134 + 23454880868215984t^136 + 26708270931980870t^138 + 30265228566674442t^140 + 34131653529364191t^142 + 38310155697571897t^144 + 42799646630124335t^146 + 47594986829348689t^148 + 52686657651574413t^150 + 58060515007372885t^152 + 63697588090508716t^154 + 69573983713758023t^156 + 75660852189950216t^158 + 81924477334761042t^160 + 88326437791529640t^162 + 94823902928741789t^164 + 101370000717364754t^166 + 107914320509349734t^168 + 114403477857306157t^170 + 120781803461021734t^172 + 126992073365574034t^174 + 132976341752172838t^176 + 138676784473885088t^178 + 144036614634734540t^180 + 149000971178042979t^182 + 153517842969741439t^184 + 157538924539179268t^186 + 161020468722831085t^188 + 163924030318565827t^190 + 166217170418719689t^192 + 167874016290695618t^194 + 168875752370039801t^196 + 169210940569431904t^198 + 168875752370039801t^200 + 167874016290695618t^202 + 166217170418719689t^204 + 163924030318565827t^206 + 161020468722831085t^208 + 157538924539179268t^210 + 153517842969741439t^212 + 149000971178042979t^214 + 144036614634734540t^216 + 138676784473885088t^218 + 132976341752172838t^220 + 126992073365574034t^222 + 120781803461021734t^224 + 114403477857306157t^226 + 107914320509349734t^228 + 101370000717364754t^230 + 94823902928741789t^232 + 88326437791529640t^234 + 81924477334761042t^236 + 75660852189950216t^238 + 69573983713758023t^240 + 63697588090508716t^242 + 58060515007372885t^244 + 52686657651574413t^246 + 47594986829348689t^248 + 42799646630124335t^250 + 38310155697571897t^252 + 34131653529364191t^254 + 30265228566674442t^256 + 26708270931980870t^258 + 23454880868215984t^260 + 20496280522892201t^262 + 17821255959669095t^264 + 15416582904745600t^266 + 13267460256461451t^268 + 11357911416698039t^270 + 9671175601669097t^272 + 8190055985475190t^274 + 6897245571573098t^276 + 5775604247640348t^278 + 4808406912033183t^280 + 3979542167320963t^282 + 3273680430050588t^284 + 2676396167205380t^286 + 2174261846638307t^288 + 1754902558254528t^290 + 1407027324715603t^292 + 1120429310910923t^294 + 885969085020306t^296 + 695535474333511t^298 + 541996086847665t^300 + 419133599770660t^302 + 321577713371961t^304 + 244729831870232t^306 + 184688242166705t^308 + 138171399515663t^310 + 102445130580609t^312 + 75251660645652t^314 + 54744574508758t^316 + 39427794406359t^318 + 28101294636548t^320 + 19811780140866t^322 + 13809982226062t^324 + 9512958483185t^326 + 6472286726974t^328 + 4346732639516t^330 + 2879776991399t^332 + 1880803164406t^334 + 1210030918912t^336 + 766230180514t^338 + 477148117122t^340 + 291908993401t^342 + 175263298262t^344 + 103147796363t^346 + 59430973423t^348 + 33473629379t^350 + 18402427436t^352 + 9856315443t^354 + 5133646455t^356 + 2593901734t^358 + 1268624162t^360 + 598609706t^362 + 271792913t^364 + 118193567t^366 + 49083610t^368 + 19325255t^370 + 7196202t^372 + 2501947t^374 + 813348t^376 + 240419t^378 + 65914t^380 + 15467t^382 + 3479t^384 + 540t^386 + 112t^388 + 2t^390 + 3t^392 + t^396) / (1 – t^4)(1 – t^6)(1 – t^8)(1 – t^10)(1 – t^12)(1 – t^14) (1 – t^16)(1 – t^18)(1 – t^20)(1 – t^22)(1 – t^24)(1 – t^26) (1 – t^28)(1 – t^30)(1 – t^32)(1 – t^34)(1 – t^36)(1 – t^38) (1 – t^40)

### n=22

(1 + 2t^4 + 18t^6 + 24t^7 + 116t^8 + 243t^9 + 717t^10 + 1572t^11 + 3924t^12 + 8340t^13 + 18470t^14 + 37625t^15 + 76630t^16 + 148208t^17 + 283514t^18 + 522331t^19 + 947836t^20 + 1671831t^21 + 2900518t^22 + 4918038t^23 + 8206738t^24 + 13430479t^25 + 21651222t^26 + 34321280t^27 + 53648607t^28 + 82622822t^29 + 125610716t^30 + 188437295t^31 + 279335249t^32 + 409120728t^33 + 592633603t^34 + 849075185t^35 + 1204121123t^36 + 1690461257t^37 + 2350823233t^38 + 3238760409t^39 + 4422798261t^40 + 5987524933t^41 + 8039091211t^42 + 10706539494t^43 + 14148906853t^44 + 18556664949t^45 + 24160525319t^46 + 31232793423t^47 + 40097907419t^48 + 51133224680t^49 + 64781263962t^50 + 81549281913t^51 + 102022871747t^52 + 126863563054t^53 + 156823307474t^54 + 192738925845t^55 + 235546969679t^56 + 286273727273t^57 + 346049600383t^58 + 416093413768t^59 + 497725504559t^60 + 592344793327t^61 + 701440045663t^62 + 826558411679t^63 + 969314210146t^64 + 1131348158013t^65 + 1314333550495t^66 + 1519925472506t^67 + 1749765031480t^68 + 2005418632072t^69 + 2288381109565t^70 + 2600006162466t^71 + 2941510148925t^72 + 3313895018628t^73 + 3717955557606t^74 + 4154197120360t^75 + 4622849106892t^76 + 5123780749727t^77 + 5656524265946t^78 + 6220191638409t^79 + 6813511389218t^80 + 7434749644541t^81 + 8081763479148t^82 + 8751929669723t^83 + 9442217262569t^84 + 10149125916876t^85 + 10868779806984t^86 + 11596876798420t^87 + 12328803599729t^88 + 13059596228810t^89 + 13784075222171t^90 + 14496815483292t^91 + 15192299311555t^92 + 15864892779327t^93 + 16509011872306t^94 + 17119101881866t^95 + 17689811241388t^96 + 18215968504415t^97 + 18692758260811t^98 + 19115690380071t^99 + 19480771058402t^100 + 19784459737578t^101 + 20023829201397t^102 + 20196505157060t^103 + 20300810784826t^104 + 20335685740440t^105 + 20300810784826t^106 + 20196505157060t^107 + 20023829201397t^108 + 19784459737578t^109 + 19480771058402t^110 + 19115690380071t^111 + 18692758260811t^112 + 18215968504415t^113 + 17689811241388t^114 + 17119101881866t^115 + 16509011872306t^116 + 15864892779327t^117 + 15192299311555t^118 + 14496815483292t^119 + 13784075222171t^120 + 13059596228810t^121 + 12328803599729t^122 + 11596876798420t^123 + 10868779806984t^124 + 10149125916876t^125 + 9442217262569t^126 + 8751929669723t^127 + 8081763479148t^128 + 7434749644541t^129 + 6813511389218t^130 + 6220191638409t^131 + 5656524265946t^132 + 5123780749727t^133 + 4622849106892t^134 + 4154197120360t^135 + 3717955557606t^136 + 3313895018628t^137 + 2941510148925t^138 + 2600006162466t^139 + 2288381109565t^140 + 2005418632072t^141 + 1749765031480t^142 + 1519925472506t^143 + 1314333550495t^144 + 1131348158013t^145 + 969314210146t^146 + 826558411679t^147 + 701440045663t^148 + 592344793327t^149 + 497725504559t^150 + 416093413768t^151 + 346049600383t^152 + 286273727273t^153 + 235546969679t^154 + 192738925845t^155 + 156823307474t^156 + 126863563054t^157 + 102022871747t^158 + 81549281913t^159 + 64781263962t^160 + 51133224680t^161 + 40097907419t^162 + 31232793423t^163 + 24160525319t^164 + 18556664949t^165 + 14148906853t^166 + 10706539494t^167 + 8039091211t^168 + 5987524933t^169 + 4422798261t^170 + 3238760409t^171 + 2350823233t^172 + 1690461257t^173 + 1204121123t^174 + 849075185t^175 + 592633603t^176 + 409120728t^177 + 279335249t^178 + 188437295t^179 + 125610716t^180 + 82622822t^181 + 53648607t^182 + 34321280t^183 + 21651222t^184 + 13430479t^185 + 8206738t^186 + 4918038t^187 + 2900518t^188 + 1671831t^189 + 947836t^190 + 522331t^191 + 283514t^192 + 148208t^193 + 76630t^194 + 37625t^195 + 18470t^196 + 8340t^197 + 3924t^198 + 1572t^199 + 717t^200 + 243t^201 + 116t^202 + 24t^203 + 18t^204 + 2t^206 + t^210) / (1 – t^2)(1 – t^4)(1 – t^5)(1 – t^6)^2(1 – t^7)(1 – t^8)(1 – t^9) (1 – t^10)(1 – t^11)(1 – t^12)(1 – t^13)(1 – t^14)(1 – t^15) (1 – t^16)(1 – t^17)(1 – t^18)(1 – t^19)(1 – t^20)(1 – t^21)

### n=23

(1 + 3t^4 + 3t^6 + 159t^8 + 918t^10 + 6546t^12 + 33342t^14 + 158010t^16 + 645900t^18 + 2419256t^20 + 8238357t^22 + 26061605t^24 + 76821539t^26 + 213251016t^28 + 559907526t^30 + 1399379788t^32 + 3342171465t^34 + 7661055690t^36 + 16907735803t^38 + 36041565842t^40 + 74394966408t^42 + 149060642023t^44 + 290501462802t^46 + 551737302703t^48 + 1022905560970t^50 + 1854077791827t^52 + 3290042334639t^54 + 5722745785945t^56 + 9768508496571t^58 + 16380524264730t^60 + 27009407964160t^62 + 43830034448654t^64 + 70056198948144t^66 + 110373001218674t^68 + 171521235544093t^70 + 263080216310789t^72 + 398500689683293t^74 + 596456264752822t^76 + 882587069066775t^78 + 1291731449152308t^80 + 1870745262340381t^82 + 2682036139314419t^84 + 3807939581365898t^86 + 5356097256390880t^88 + 7465989045793980t^90 + 10316809017908588t^92 + 14136852815757879t^94 + 19214626738233075t^96 + 25911845101010902t^98 + 34678528344262872t^100 + 46070341170144039t^102 + 60768358272717461t^104 + 79601333164410532t^106 + 103570596752130785t^108 + 133877551788549055t^110 + 171953785995740218t^112 + 219493608752466017t^114 + 278488884393261581t^116 + 351265752682071039t^118 + 440522917070058071t^120 + 549370829702976107t^122 + 681371229901015722t^124 + 840576070736901297t^126 + 1031565055542783054t^128 + 1259480515853012284t^130 + 1530058636478327782t^132 + 1849655480107346036t^134 + 2225266657482756021t^136 + 2664538879936967348t^138 + 3175772179787757365t^140 + 3767910925661537804t^142 + 4450522499684015550t^144 + 5233761799692010718t^146 + 6128320694049712427t^148 + 7145360805228552210t^150 + 8296429214409697286t^152 + 9593355868908289756t^154 + 11048132959413403976t^156 + 12672775641396264581t^158 + 14479165234488790997t^160 + 16478875022362863484t^162 + 18682980800724608869t^164 + 21101857141499058204t^166 + 23744962609771705566t^168 + 26620615761213233217t^170 + 29735766226012195941t^172 + 33095763478567230975t^174 + 36704128539237605978t^176 + 40562331779301092195t^178 + 44669582773084996242t^180 + 49022635632334505848t^182 + 53615616118312343426t^184 + 58439873831670852613t^186 + 63483865701495687100t^188 + 68733073480084926660t^190 + 74169960922990346682t^192 + 79773972292514936325t^194 + 85521576859839991181t^196 + 91386359535213168408t^198 + 97339160892235767154t^200 + 103348264854171273765t^202 + 109379635603604614571t^204 + 115397199905290669280t^206 + 121363174556359622373t^208 + 127238433032874876351t^210 + 132982909240129587890t^212 + 138556030466874217547t^214 + 143917175878194646658t^216 + 149026151025248544038t^218 + 153843673550969344328t^220 + 158331859467340115959t^222 + 162454704594697336173t^224 + 166178550084809298267t^226 + 169472526694198269131t^228 + 172308966993229307719t^230 + 174663780949746829939t^232 + 176516785052202257892t^234 + 177851981838423425969t^236 + 178657781614856536370t^238 + 178927165202855442572t^240 + 178657781614856536370t^242 + 177851981838423425969t^244 + 176516785052202257892t^246 + 174663780949746829939t^248 + 172308966993229307719t^250 + 169472526694198269131t^252 + 166178550084809298267t^254 + 162454704594697336173t^256 + 158331859467340115959t^258 + 153843673550969344328t^260 + 149026151025248544038t^262 + 143917175878194646658t^264 + 138556030466874217547t^266 + 132982909240129587890t^268 + 127238433032874876351t^270 + 121363174556359622373t^272 + 115397199905290669280t^274 + 109379635603604614571t^276 + 103348264854171273765t^278 + 97339160892235767154t^280 + 91386359535213168408t^282 + 85521576859839991181t^284 + 79773972292514936325t^286 + 74169960922990346682t^288 + 68733073480084926660t^290 + 63483865701495687100t^292 + 58439873831670852613t^294 + 53615616118312343426t^296 + 49022635632334505848t^298 + 44669582773084996242t^300 + 40562331779301092195t^302 + 36704128539237605978t^304 + 33095763478567230975t^306 + 29735766226012195941t^308 + 26620615761213233217t^310 + 23744962609771705566t^312 + 21101857141499058204t^314 + 18682980800724608869t^316 + 16478875022362863484t^318 + 14479165234488790997t^320 + 12672775641396264581t^322 + 11048132959413403976t^324 + 9593355868908289756t^326 + 8296429214409697286t^328 + 7145360805228552210t^330 + 6128320694049712427t^332 + 5233761799692010718t^334 + 4450522499684015550t^336 + 3767910925661537804t^338 + 3175772179787757365t^340 + 2664538879936967348t^342 + 2225266657482756021t^344 + 1849655480107346036t^346 + 1530058636478327782t^348 + 1259480515853012284t^350 + 1031565055542783054t^352 + 840576070736901297t^354 + 681371229901015722t^356 + 549370829702976107t^358 + 440522917070058071t^360 + 351265752682071039t^362 + 278488884393261581t^364 + 219493608752466017t^366 + 171953785995740218t^368 + 133877551788549055t^370 + 103570596752130785t^372 + 79601333164410532t^374 + 60768358272717461t^376 + 46070341170144039t^378 + 34678528344262872t^380 + 25911845101010902t^382 + 19214626738233075t^384 + 14136852815757879t^386 + 10316809017908588t^388 + 7465989045793980t^390 + 5356097256390880t^392 + 3807939581365898t^394 + 2682036139314419t^396 + 1870745262340381t^398 + 1291731449152308t^400 + 882587069066775t^402 + 596456264752822t^404 + 398500689683293t^406 + 263080216310789t^408 + 171521235544093t^410 + 110373001218674t^412 + 70056198948144t^414 + 43830034448654t^416 + 27009407964160t^418 + 16380524264730t^420 + 9768508496571t^422 + 5722745785945t^424 + 3290042334639t^426 + 1854077791827t^428 + 1022905560970t^430 + 551737302703t^432 + 290501462802t^434 + 149060642023t^436 + 74394966408t^438 + 36041565842t^440 + 16907735803t^442 + 7661055690t^444 + 3342171465t^446 + 1399379788t^448 + 559907526t^450 + 213251016t^452 + 76821539t^454 + 26061605t^456 + 8238357t^458 + 2419256t^460 + 645900t^462 + 158010t^464 + 33342t^466 + 6546t^468 + 918t^470 + 159t^472 + 3t^474 + 3t^476 + t^480) / (1 – t^4)(1 – t^6)(1 – t^8)(1 – t^10)(1 – t^12)(1 – t^14)(1 – t^16) (1 – t^18)(1 – t^20)(1 – t^22)(1 – t^24)(1 – t^26)(1 – t^28) (1 – t^30)(1 – t^32)(1 – t^34)(1 – t^36)(1 – t^38)(1 – t^40) (1 – t^42)(1 – t^44)

### n=24

(1 + 3t^4 + 5t^5 + 22t^6 + 50t^7 + 161t^8 + 410t^9 + 1140t^10 + 2808t^11 + 6991t^12 + 16199t^13 + 36859t^14 + 80010t^15 + 169421t^16 + 346121t^17 + 689947t^18 + 1336028t^19 + 2528528t^20 + 4670438t^21 + 8449357t^22 + 14968148t^23 + 26025211t^24 + 44423184t^25 + 74560924t^26 + 123110049t^27 + 200201862t^28 + 320813495t^29 + 507041603t^30 + 790779399t^31 + 1217881983t^32 + 1853082547t^33 + 2787305828t^34 + 4146285473t^35 + 6102914802t^36 + 8891714037t^37 + 12828922109t^38 + 18335849747t^39 + 25970411969t^40 + 36463444967t^41 + 50766544654t^42 + 70106566677t^43 + 96055848819t^44 + 130611273929t^45 + 176294077526t^46 + 236260806268t^47 + 314440780906t^48 + 415686796764t^49 + 545958588510t^50 + 712520954002t^51 + 924180944791t^52 + 1191539827621t^53 + 1527289937061t^54 + 1946524208144t^55 + 2467095245250t^56 + 3109981870291t^57 + 3899707778226t^58 + 4864758338084t^59 + 6038049238675t^60 + 7457378700401t^61 + 9165927715226t^62 + 11212723264911t^63 + 13653141566979t^64 + 16549347183387t^65 + 19970759966163t^66 + 23994424008053t^67 + 28705388495679t^68 + 34196950655128t^69 + 40570891843897t^70 + 47937531085658t^71 + 56415752168625t^72 + 66132800675574t^73 + 77224036793196t^74 + 89832410691882t^75 + 104107880721344t^76 + 120206510443320t^77 + 138289504277080t^78 + 158521885428959t^79 + 181071120920863t^80 + 206105363625597t^81 + 233791665949818t^82 + 264293800024765t^83 + 297770093432862t^84 + 334370877999768t^85 + 374236019258930t^86 + 417492084225375t^87 + 464249676150170t^88 + 514600451190458t^89 + 568614408301291t^90 + 626336920289549t^91 + 687786160642371t^92 + 752950342462258t^93 + 821785485884455t^94 + 894213074068083t^95 + 970118373456853t^96 + 1049348716366855t^97 + 1131712577949459t^98 + 1216978678300190t^99 + 1304875993404447t^100 + 1395093834298654t^101 + 1487282925178084t^102 + 1581056564322066t^103 + 1675992841680187t^104 + 1771636919407880t^105 + 1867504387728908t^106 + 1963084625347838t^107 + 2057845212109979t^108 + 2151236247650709t^109 + 2242695657576844t^110 + 2331654270014146t^111 + 2417541776323760t^112 + 2499792295577520t^113 + 2577850688959356t^114 + 2651178288955232t^115 + 2719259223507973t^116 + 2781605956195677t^117 + 2837765257346956t^118 + 2887323196198862t^119 + 2929910405074852t^120 + 2965206186731099t^121 + 2992942753356401t^122 + 3012908161933130t^123 + 3024949270785865t^124 + 3028973288002032t^125 + 3024949270785865t^126 + 3012908161933130t^127 + 2992942753356401t^128 + 2965206186731099t^129 + 2929910405074852t^130 + 2887323196198862t^131 + 2837765257346956t^132 + 2781605956195677t^133 + 2719259223507973t^134 + 2651178288955232t^135 + 2577850688959356t^136 + 2499792295577520t^137 + 2417541776323760t^138 + 2331654270014146t^139 + 2242695657576844t^140 + 2151236247650709t^141 + 2057845212109979t^142 + 1963084625347838t^143 + 1867504387728908t^144 + 1771636919407880t^145 + 1675992841680187t^146 + 1581056564322066t^147 + 1487282925178084t^148 + 1395093834298654t^149 + 1304875993404447t^150 + 1216978678300190t^151 + 1131712577949459t^152 + 1049348716366855t^153 + 970118373456853t^154 + 894213074068083t^155 + 821785485884455t^156 + 752950342462258t^157 + 687786160642371t^158 + 626336920289549t^159 + 568614408301291t^160 + 514600451190458t^161 + 464249676150170t^162 + 417492084225375t^163 + 374236019258930t^164 + 334370877999768t^165 + 297770093432862t^166 + 264293800024765t^167 + 233791665949818t^168 + 206105363625597t^169 + 181071120920863t^170 + 158521885428959t^171 + 138289504277080t^172 + 120206510443320t^173 + 104107880721344t^174 + 89832410691882t^175 + 77224036793196t^176 + 66132800675574t^177 + 56415752168625t^178 + 47937531085658t^179 + 40570891843897t^180 + 34196950655128t^181 + 28705388495679t^182 + 23994424008053t^183 + 19970759966163t^184 + 16549347183387t^185 + 13653141566979t^186 + 11212723264911t^187 + 9165927715226t^188 + 7457378700401t^189 + 6038049238675t^190 + 4864758338084t^191 + 3899707778226t^192 + 3109981870291t^193 + 2467095245250t^194 + 1946524208144t^195 + 1527289937061t^196 + 1191539827621t^197 + 924180944791t^198 + 712520954002t^199 + 545958588510t^200 + 415686796764t^201 + 314440780906t^202 + 236260806268t^203 + 176294077526t^204 + 130611273929t^205 + 96055848819t^206 + 70106566677t^207 + 50766544654t^208 + 36463444967t^209 + 25970411969t^210 + 18335849747t^211 + 12828922109t^212 + 8891714037t^213 + 6102914802t^214 + 4146285473t^215 + 2787305828t^216 + 1853082547t^217 + 1217881983t^218 + 790779399t^219 + 507041603t^220 + 320813495t^221 + 200201862t^222 + 123110049t^223 + 74560924t^224 + 44423184t^225 + 26025211t^226 + 14968148t^227 + 8449357t^228 + 4670438t^229 + 2528528t^230 + 1336028t^231 + 689947t^232 + 346121t^233 + 169421t^234 + 80010t^235 + 36859t^236 + 16199t^237 + 6991t^238 + 2808t^239 + 1140t^240 + 410t^241 + 161t^242 + 50t^243 + 22t^244 + 5t^245 + 3t^246 + t^250) / (1 – t^2)(1 – t^3)(1 – t^4) (1 – t^5)(1 – t^6)(1 – t^7)(1 – t^8)(1 – t^9)(1 – t^10)(1 – t^11) (1 – t^12)(1 – t^13)(1 – t^14)(1 – t^15)(1 – t^16)(1 – t^17) (1 – t^18)(1 – t^19)(1 – t^20)(1 – t^21)(1 – t^22)(1 – t^23)

= 1 + t^2 + t^3 + 5t^4 + 7t^5 + 29t^6 + 62t^7 + 201t^8 + 506t^9 + 1429t^10 + 3569t^11 + 9113t^12 + 21660t^13 + 50866t^14 + 114049t^15 + 250256t^16 + 530471t^17 + 1099354t^18 + 2215994t^19 + 4372347t^20 + ...

### n=25

(1 + 3t^4 + 5t^6 + 222t^8 + 1502t^10 + 11821t^12 + 68160t^14 + 357762t^16 + 1625860t^18 + 6709808t^20 + 25138116t^22 + 87046050t^24 + 280160016t^26 + 846121749t^28 + 2410944563t^30 + 6520970770t^32 + 16815721636t^34 + 41521184276t^36 + 98508905536t^38 + 225286549371t^40 + 498005650572t^42 + 1066743966417t^44 + 2219015152119t^46 + 4491580172716t^48 + 8862251570482t^50 + 17072435065954t^52 + 32157541208126t^54 + 59303768036561t^56 + 107205351380220t^58 + 190180199531403t^60 + 331411329546026t^62 + 567841090141965t^64 + 957448072224676t^66 + 1589923554447163t^68 + 2602119925219313t^70 + 4200133095076674t^72 + 6690472530288153t^74 + 10523567771409358t^76 + 16353797601582608t^78 + 25121460991405799t^80 + 38163525979011174t^82 + 57361820580998838t^84 + 85339376425536255t^86 + 125718229681185986t^88 + 183454807985267846t^90 + 265272555157753420t^92 + 380215164524726737t^94 + 540348366711047137t^96 + 761642877622796151t^98 + 1065076777391577246t^100 + 1478001114414834236t^102 + 2035819183439021111t^104 + 2784036053715741981t^106 + 3780742271542488808t^108 + 5099601889639112351t^110 + 6833422471453673036t^112 + 9098390250647782554t^114 + 12039060433877592957t^116 + 15834196349368709806t^118 + 20703556128912269172t^120 + 26915726064923581614t^122 + 34797101543106716592t^124 + 44742111934164906645t^126 + 57224782773618774934t^128 + 72811717203506813354t^130 + 92176569159558677303t^132 + 116116060745867367995t^134 + 145567578867341713600t^136 + 181628355870647167958t^138 + 225576212689152721084t^140 + 278891801340173468813t^142 + 343282248230958076702t^144 + 420706046555079286350t^146 + 513399002339077598737t^148 + 623900974942838111421t^150 + 755083102906624423921t^152 + 910175134030667238138t^154 + 1092792426142939145297t^156 + 1306962108599000662657t^158 + 1557147844865842703463t^160 + 1848272561063142373374t^162 + 2185738465110111504612t^164 + 2575443613194343407584t^166 + 3023794257393804293464t^168 + 3537712155351168811454t^170 + 4124636025826823039791t^172 + 4792516303668913594924t^174 + 5549802384795146130342t^176 + 6405421551139595356612t^178 + 7368748843953168319612t^180 + 8449567188251902956514t^182 + 9658017196210751716400t^184 + 11004536149562073937103t^186 + 12499785834779253137936t^188 + 14154569014253379107699t^190 + 15979734537387205588838t^192 + 17986071237228340265218t^194 + 20184191019816158179340t^196 + 22584401717912119943843t^198 + 25196570572391412515283t^200 + 28029979377881058345110t^202 + 31093172635785006456171t^204 + 34393800222019419144402t^206 + 37938456379661683694264t^208 + 41732516980978134724890t^210 + 45779977281681403798492t^212 + 50083292472525795530416t^214 + 54643223565985084081007t^216 + 59458691164584764254996t^218 + 64526639824719301756894t^220 + 69841915646577840724578t^222 + 75397159806851132049166t^224 + 81182720562034436644567t^226 + 87186586245008444588256t^228 + 93394341476366230374218t^230 + 99789148713235449473107t^232 + 106351756846440012355171t^234 + 113060538370822203225204t^236 + 119891556141762593230539t^238 + 126818660472935748249363t^240 + 133813616738565309279154t^242 + 140846263337321312915383t^244 + 147884699231229808107014t^246 + 154895499949661391619533t^248 + 161843960286932760430022t^250 + 168694361617325305219966t^252 + 175410261109880720025356t^254 + 181954799876821295643391t^256 + 188291026506373871657912t^258 + 194382232274666740724282t^260 + 200192293841977828987952t^262 + 205686019206213030106415t^264 + 210829492319747119779766t^266 + 215590411890350409469955t^268 + 219938419662553616597569t^270 + 223845413750077305465619t^272 + 227285842516310687657368t^274 + 230236974934239798265180t^276 + 232679143430138999564803t^278 + 234595955797574738753093t^280 + 235974472971067329689260t^282 + 236805350154344993711278t^284 + 237082939102855202614432t^286 + 236805350154344993711278t^288 + 235974472971067329689260t^290 + 234595955797574738753093t^292 + 232679143430138999564803t^294 + 230236974934239798265180t^296 + 227285842516310687657368t^298 + 223845413750077305465619t^300 + 219938419662553616597569t^302 + 215590411890350409469955t^304 + 210829492319747119779766t^306 + 205686019206213030106415t^308 + 200192293841977828987952t^310 + 194382232274666740724282t^312 + 188291026506373871657912t^314 + 181954799876821295643391t^316 + 175410261109880720025356t^318 + 168694361617325305219966t^320 + 161843960286932760430022t^322 + 154895499949661391619533t^324 + 147884699231229808107014t^326 + 140846263337321312915383t^328 + 133813616738565309279154t^330 + 126818660472935748249363t^332 + 119891556141762593230539t^334 + 113060538370822203225204t^336 + 106351756846440012355171t^338 + 99789148713235449473107t^340 + 93394341476366230374218t^342 + 87186586245008444588256t^344 + 81182720562034436644567t^346 + 75397159806851132049166t^348 + 69841915646577840724578t^350 + 64526639824719301756894t^352 + 59458691164584764254996t^354 + 54643223565985084081007t^356 + 50083292472525795530416t^358 + 45779977281681403798492t^360 + 41732516980978134724890t^362 + 37938456379661683694264t^364 + 34393800222019419144402t^366 + 31093172635785006456171t^368 + 28029979377881058345110t^370 + 25196570572391412515283t^372 + 22584401717912119943843t^374 + 20184191019816158179340t^376 + 17986071237228340265218t^378 + 15979734537387205588838t^380 + 14154569014253379107699t^382 + 12499785834779253137936t^384 + 11004536149562073937103t^386 + 9658017196210751716400t^388 + 8449567188251902956514t^390 + 7368748843953168319612t^392 + 6405421551139595356612t^394 + 5549802384795146130342t^396 + 4792516303668913594924t^398 + 4124636025826823039791t^400 + 3537712155351168811454t^402 + 3023794257393804293464t^404 + 2575443613194343407584t^406 + 2185738465110111504612t^408 + 1848272561063142373374t^410 + 1557147844865842703463t^412 + 1306962108599000662657t^414 + 1092792426142939145297t^416 + 910175134030667238138t^418 + 755083102906624423921t^420 + 623900974942838111421t^422 + 513399002339077598737t^424 + 420706046555079286350t^426 + 343282248230958076702t^428 + 278891801340173468813t^430 + 225576212689152721084t^432 + 181628355870647167958t^434 + 145567578867341713600t^436 + 116116060745867367995t^438 + 92176569159558677303t^440 + 72811717203506813354t^442 + 57224782773618774934t^444 + 44742111934164906645t^446 + 34797101543106716592t^448 + 26915726064923581614t^450 + 20703556128912269172t^452 + 15834196349368709806t^454 + 12039060433877592957t^456 + 9098390250647782554t^458 + 6833422471453673036t^460 + 5099601889639112351t^462 + 3780742271542488808t^464 + 2784036053715741981t^466 + 2035819183439021111t^468 + 1478001114414834236t^470 + 1065076777391577246t^472 + 761642877622796151t^474 + 540348366711047137t^476 + 380215164524726737t^478 + 265272555157753420t^480 + 183454807985267846t^482 + 125718229681185986t^484 + 85339376425536255t^486 + 57361820580998838t^488 + 38163525979011174t^490 + 25121460991405799t^492 + 16353797601582608t^494 + 10523567771409358t^496 + 6690472530288153t^498 + 4200133095076674t^500 + 2602119925219313t^502 + 1589923554447163t^504 + 957448072224676t^506 + 567841090141965t^508 + 331411329546026t^510 + 190180199531403t^512 + 107205351380220t^514 + 59303768036561t^516 + 32157541208126t^518 + 17072435065954t^520 + 8862251570482t^522 + 4491580172716t^524 + 2219015152119t^526 + 1066743966417t^528 + 498005650572t^530 + 225286549371t^532 + 98508905536t^534 + 41521184276t^536 + 16815721636t^538 + 6520970770t^540 + 2410944563t^542 + 846121749t^544 + 280160016t^546 + 87046050t^548 + 25138116t^550 + 6709808t^552 + 1625860t^554 + 357762t^556 + 68160t^558 + 11821t^560 + 1502t^562 + 222t^564 + 5t^566 + 3t^568 + t^572) / (1 – t^4)(1 – t^6)(1 – t^8) (1 – t^10)(1 – t^12)(1 – t^14)(1 – t^16)(1 – t^18)(1 – t^20) (1 – t^22)(1 – t^24)(1 – t^26)(1 – t^28)(1 – t^30)(1 – t^32) (1 – t^34)(1 – t^36)(1 – t^38)(1 – t^40)(1 – t^42)(1 – t^44) (1 – t^46)(1 – t^48)

### n=26

(1 + 3t^4 + t^5 + 27t^6 + 49t^7 + 233t^8 + 589t^9 + 1909t^10 + 4901t^11 + 13458t^12 + 32968t^13 + 81006t^14 + 186793t^15 + 423359t^16 + 919195t^17 + 1952674t^18 + 4014960t^19 + 8074868t^20 + 15816492t^21 + 30341435t^22 + 56915164t^23 + 104742314t^24 + 189033656t^25 + 335274531t^26 + 584464376t^27 + 1002868167t^28 + 1694360168t^29 + 2821685248t^30 + 4633757103t^31 + 7509982218t^32 + 12017459429t^33 + 18999508744t^34 + 29689875143t^35 + 45882665322t^36 + 70150555529t^37 + 106159012487t^38 + 159066938374t^39 + 236087456201t^40 + 347196945245t^41 + 506103167131t^42 + 731457710415t^43 + 1048475592845t^44 + 1490943178507t^45 + 2103847750051t^46 + 2946603872714t^47 + 4097203010694t^48 + 5657251052276t^49 + 7758340735177t^50 + 10569705022830t^51 + 14307749726848t^52 + 19247379977450t^53 + 25735903889804t^54 + 34209374975003t^55 + 45212378411620t^56 + 59421036708832t^57 + 77670497705625t^58 + 100986548082595t^59 + 130622907282512t^60 + 168103646506476t^61 + 215272611871287t^62 + 274349008883484t^63 + 347991376748847t^64 + 439368709603749t^65 + 552241320815448t^66 + 691049664530054t^67 + 861014092862770t^68 + 1068243047255884t^69 + 1319853052140060t^70 + 1624097096457530t^71 + 1990505166783979t^72 + 2430032379940646t^73 + 2955218890326103t^74 + 3580355638504188t^75 + 4321660554817331t^76 + 5197457656483251t^77 + 6228364142486577t^78 + 7437476049408678t^79 + 8850558161371944t^80 + 10496226631033559t^81 + 12406130741483473t^82 + 14615119969698822t^83 + 17161403732824446t^84 + 20086687533693819t^85 + 23436294143555276t^86 + 27259251004270279t^87 + 31608354133575464t^88 + 36540187171324169t^89 + 42115107975544943t^90 + 48397178915758422t^91 + 55454055979039256t^92 + 63356810463387367t^93 + 72179701729884386t^94 + 81999872600969429t^95 + 92896989932053147t^96 + 104952799964488066t^97 + 118250625781636843t^98 + 132874774701732503t^99 + 148909888435017470t^100 + 166440202252834224t^101 + 185548752145331362t^102 + 206316494729980278t^103 + 228821385589890131t^104 + 253137379296074060t^105 + 279333403876954182t^106 + 307472271326905816t^107 + 337609584176846770t^108 + 369792597703959590t^109 + 404059105033073894t^110 + 440436302131147643t^111 + 478939706883743697t^112 + 519572085945109539t^113 + 562322469938600837t^114 + 607165206455894093t^115 + 654059137075980625t^116 + 702946842521892441t^117 + 753754046875233019t^118 + 806389118494813001t^119 + 860742762178542468t^120 + 916687832571158871t^121 + 974079365882365196t^122 + 1032754751163597682t^123 + 1092534139656164138t^124 + 1153221003796562424t^125 + 1214602944872854503t^126 + 1276452650581050929t^127 + 1338529101235486768t^128 + 1400578915189843955t^129 + 1462337931516205497t^130 + 1523532909887543313t^131 + 1583883444847297418t^132 + 1643103964290316629t^133 + 1700905908710201102t^134 + 1756999951827334273t^135 + 1811098359078026693t^136 + 1862917336738358506t^137 + 1912179468928055650t^138 + 1958616089127882804t^139 + 2001969685343536797t^140 + 2041996181358982183t^141 + 2078467196341906763t^142 + 2111172123209958297t^143 + 2139920132404221468t^144 + 2164541941610267041t^145 + 2184891463672853518t^146 + 2200847175424831718t^147 + 2212313326261324552t^148 + 2219220833238661594t^149 + 2221527988763778294t^150 + 2219220833238661594t^151 + 2212313326261324552t^152 + 2200847175424831718t^153 + 2184891463672853518t^154 + 2164541941610267041t^155 + 2139920132404221468t^156 + 2111172123209958297t^157 + 2078467196341906763t^158 + 2041996181358982183t^159 + 2001969685343536797t^160 + 1958616089127882804t^161 + 1912179468928055650t^162 + 1862917336738358506t^163 + 1811098359078026693t^164 + 1756999951827334273t^165 + 1700905908710201102t^166 + 1643103964290316629t^167 + 1583883444847297418t^168 + 1523532909887543313t^169 + 1462337931516205497t^170 + 1400578915189843955t^171 + 1338529101235486768t^172 + 1276452650581050929t^173 + 1214602944872854503t^174 + 1153221003796562424t^175 + 1092534139656164138t^176 + 1032754751163597682t^177 + 974079365882365196t^178 + 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12406130741483473t^218 + 10496226631033559t^219 + 8850558161371944t^220 + 7437476049408678t^221 + 6228364142486577t^222 + 5197457656483251t^223 + 4321660554817331t^224 + 3580355638504188t^225 + 2955218890326103t^226 + 2430032379940646t^227 + 1990505166783979t^228 + 1624097096457530t^229 + 1319853052140060t^230 + 1068243047255884t^231 + 861014092862770t^232 + 691049664530054t^233 + 552241320815448t^234 + 439368709603749t^235 + 347991376748847t^236 + 274349008883484t^237 + 215272611871287t^238 + 168103646506476t^239 + 130622907282512t^240 + 100986548082595t^241 + 77670497705625t^242 + 59421036708832t^243 + 45212378411620t^244 + 34209374975003t^245 + 25735903889804t^246 + 19247379977450t^247 + 14307749726848t^248 + 10569705022830t^249 + 7758340735177t^250 + 5657251052276t^251 + 4097203010694t^252 + 2946603872714t^253 + 2103847750051t^254 + 1490943178507t^255 + 1048475592845t^256 + 731457710415t^257 + 506103167131t^258 + 347196945245t^259 + 236087456201t^260 + 159066938374t^261 + 106159012487t^262 + 70150555529t^263 + 45882665322t^264 + 29689875143t^265 + 18999508744t^266 + 12017459429t^267 + 7509982218t^268 + 4633757103t^269 + 2821685248t^270 + 1694360168t^271 + 1002868167t^272 + 584464376t^273 + 335274531t^274 + 189033656t^275 + 104742314t^276 + 56915164t^277 + 30341435t^278 + 15816492t^279 + 8074868t^280 + 4014960t^281 + 1952674t^282 + 919195t^283 + 423359t^284 + 186793t^285 + 81006t^286 + 32968t^287 + 13458t^288 + 4901t^289 + 1909t^290 + 589t^291 + 233t^292 + 49t^293 + 27t^294 + t^295 + 3t^296 + t^300) / (1 – t^2)(1 – t^4)(1 – t^5)(1 – t^6)^2(1 – t^7)(1 – t^8)(1 – t^9) (1 – t^10)(1 – t^11)(1 – t^12)(1 – t^13)(1 – t^14)(1 – t^15) (1 – t^16)(1 – t^17)(1 – t^18)(1 – t^19)(1 – t^20)(1 – t^21) (1 – t^22)(1 – t^23)(1 – t^24)(1 – t^25)

### n=27

(1 + 4t^4 + 7t^6 + 303t^8 + 2389t^10 + 20581t^12 + 133110t^14 + 770750t^16 + 3868422t^18 + 17511308t^20 + 71807229t^22 + 271002608t^24 + 948161903t^26 + 3103122292t^28 + 9557939169t^30 + 27873452833t^32 + 77326930035t^34 + 204965001763t^36 + 520987710384t^38 + 1274127907247t^40 + 3006658998689t^42 + 6863823940320t^44 + 15193363263174t^46 + 32677350992181t^48 + 68414874826818t^50 + 139667809189396t^52 + 278448951689194t^54 + 542877257557468t^56 + 1036368487676655t^58 + 1939487933008614t^60 + 3561903649540405t^62 + 6425763233815655t^64 + 11397421050161966t^66 + 19892533118509333t^68 + 34190901088492001t^70 + 57913416974057671t^72 + 96735351370221342t^74 + 159440244352843028t^76 + 259458883267524269t^78 + 417091773574281667t^80 + 662684264246916957t^82 + 1041116368589139852t^84 + 1618088812200190729t^86 + 2488840647903057295t^88 + 3790127992035868022t^90 + 5716538850414503756t^92 + 8542523414678871339t^94 + 12651897169774842571t^96 + 18577034998305300954t^98 + 27050537079285957498t^100 + 39072822001676413948t^102 + 55999912731807939274t^104 + 79656636732426326508t^106 + 112481590956544332649t^108 + 157711532852482756423t^110 + 219614382579433465600t^112 + 303781760564795196063t^114 + 417493974404025151212t^116 + 570172600386502426911t^118 + 773938315655394009943t^120 + 1044294400666903336125t^122 + 1400959386354156251571t^124 + 1868875614968041118576t^126 + 2479424054294354847396t^128 + 3271879466682863969958t^130 + 4295144020044503081985t^132 + 5609801508414827531517t^134 + 7290538554041989055665t^136 + 9428983307090850517814t^138 + 12137016270028305023147t^140 + 15550611701916318155297t^142 + 19834271632980682772320t^144 + 25186117517135138773247t^146 + 31843707011462304305423t^148 + 40090644889214333711233t^150 + 50264057735647787878666t^152 + 62763001348059444947599t^154 + 78057867780742290130851t^156 + 96700855156277189036432t^158 + 119337557903313380206059t^160 + 146719727303170111215354t^162 + 179719242446317631062722t^164 + 219343319156215857234866t^166 + 266750969607082736712191t^168 + 323270707407069403480992t^170 + 390419472528033151276917t^172 + 469922726728929774725793t^174 + 563735644001708822212117t^176 + 674065291089365687870015t^178 + 803393661575229393973365t^180 + 954501392372280499627460t^182 + 1130491955328389906679001t^184 + 1334816077975284589441633t^186 + 1571296108302483178889357t^188 + 1844149997588424141911133t^190 + 2158014535368957159139803t^192 + 2517967430167536062851876t^194 + 2929547791764411570812983t^196 + 3398774533919727590508139t^198 + 3932162184228491716325018t^200 + 4536733558236587045737776t^202 + 5220028732253872850187817t^204 + 5990109731099382851868354t^206 + 6855560337981404438202264t^208 + 7825480430927523993336012t^210 + 8909474258791507393100613t^212 + 10117632086238774190180483t^214 + 11460504666843916021554117t^216 + 12949070042012013172811873t^218 + 14594692216851828731003090t^220 + 16409071326836015762067950t^222 + 18404184987436481113431795t^224 + 20592220606183095455978636t^226 + 22985498539462972653064312t^228 + 25596386086790361878613786t^230 + 28437202440292064290446622t^232 + 31520114837274764094008972t^234 + 34857026306428502275994607t^236 + 38459455542485326516106574t^238 + 42338409597942796463858450t^240 + 46504250231278720405451100t^242 + 50966554907498080578309659t^244 + 55733973594889739654345330t^246 + 60814082650903763292316977t^248 + 66213237224722565212970370t^250 + 71936423734788611957499020t^252 + 77987114089637946079150673t^254 + 84367123421545839193676517t^256 + 91076473176988932593157414t^258 + 98113261468985385934303164t^260 + 105473542625345248404401191t^262 + 113151217878778330217794398t^264 + 121137939120290440126268319t^266 + 129423027593441319313637116t^268 + 137993409323901816596652689t^270 + 146833568975687981992249109t^272 + 155925523682100374282570879t^274 + 165248818237517729961736041t^276 + 174780542835071289006260472t^278 + 184495374319087199577624594t^280 + 194365641668980425266709726t^282 + 204361416169681158638312611t^284 + 214450626431228873955630792t^286 + 224599198125860002643071825t^288 + 234771217992766203799956043t^290 + 244929121349784965484620936t^292 + 255033902024890889767096708t^294 + 265045343311347172125542930t^296 + 274922268235047489291177879t^298 + 284622807135209115659679913t^300 + 294104680275179685920659515t^302 + 303325492953976034133689343t^304 + 312243040354713104750671058t^306 + 320815619179160873038727267t^308 + 329002342949858406419635167t^310 + 336763457748532304103425820t^312 + 344060655071499120879074408t^314 + 350857378455082607387017830t^316 + 357119120524103255122820073t^318 + 362813707179494209809889048t^320 + 367911565732042687300151540t^322 + 372385973942726074135584281t^324 + 376213287108875114028759286t^326 + 379373140573207109571858844t^328 + 381848625290376009951145271t^330 + 383626434396017857683657766t^332 + 384696979045358437776434585t^334 + 385054472154512243287759444t^336 + 384696979045358437776434585t^338 + 383626434396017857683657766t^340 + 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155925523682100374282570879t^398 + 146833568975687981992249109t^400 + 137993409323901816596652689t^402 + 129423027593441319313637116t^404 + 121137939120290440126268319t^406 + 113151217878778330217794398t^408 + 105473542625345248404401191t^410 + 98113261468985385934303164t^412 + 91076473176988932593157414t^414 + 84367123421545839193676517t^416 + 77987114089637946079150673t^418 + 71936423734788611957499020t^420 + 66213237224722565212970370t^422 + 60814082650903763292316977t^424 + 55733973594889739654345330t^426 + 50966554907498080578309659t^428 + 46504250231278720405451100t^430 + 42338409597942796463858450t^432 + 38459455542485326516106574t^434 + 34857026306428502275994607t^436 + 31520114837274764094008972t^438 + 28437202440292064290446622t^440 + 25596386086790361878613786t^442 + 22985498539462972653064312t^444 + 20592220606183095455978636t^446 + 18404184987436481113431795t^448 + 16409071326836015762067950t^450 + 14594692216851828731003090t^452 + 12949070042012013172811873t^454 + 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78057867780742290130851t^516 + 62763001348059444947599t^518 + 50264057735647787878666t^520 + 40090644889214333711233t^522 + 31843707011462304305423t^524 + 25186117517135138773247t^526 + 19834271632980682772320t^528 + 15550611701916318155297t^530 + 12137016270028305023147t^532 + 9428983307090850517814t^534 + 7290538554041989055665t^536 + 5609801508414827531517t^538 + 4295144020044503081985t^540 + 3271879466682863969958t^542 + 2479424054294354847396t^544 + 1868875614968041118576t^546 + 1400959386354156251571t^548 + 1044294400666903336125t^550 + 773938315655394009943t^552 + 570172600386502426911t^554 + 417493974404025151212t^556 + 303781760564795196063t^558 + 219614382579433465600t^560 + 157711532852482756423t^562 + 112481590956544332649t^564 + 79656636732426326508t^566 + 55999912731807939274t^568 + 39072822001676413948t^570 + 27050537079285957498t^572 + 18577034998305300954t^574 + 12651897169774842571t^576 + 8542523414678871339t^578 + 5716538850414503756t^580 + 3790127992035868022t^582 + 2488840647903057295t^584 + 1618088812200190729t^586 + 1041116368589139852t^588 + 662684264246916957t^590 + 417091773574281667t^592 + 259458883267524269t^594 + 159440244352843028t^596 + 96735351370221342t^598 + 57913416974057671t^600 + 34190901088492001t^602 + 19892533118509333t^604 + 11397421050161966t^606 + 6425763233815655t^608 + 3561903649540405t^610 + 1939487933008614t^612 + 1036368487676655t^614 + 542877257557468t^616 + 278448951689194t^618 + 139667809189396t^620 + 68414874826818t^622 + 32677350992181t^624 + 15193363263174t^626 + 6863823940320t^628 + 3006658998689t^630 + 1274127907247t^632 + 520987710384t^634 + 204965001763t^636 + 77326930035t^638 + 27873452833t^640 + 9557939169t^642 + 3103122292t^644 + 948161903t^646 + 271002608t^648 + 71807229t^650 + 17511308t^652 + 3868422t^654 + 770750t^656 + 133110t^658 + 20581t^660 + 2389t^662 + 303t^664 + 7t^666 + 4t^668 + t^672) / (1 – t^4)(1 – t^6)(1 – t^8)(1 – t^10)(1 – t^12)(1 – t^14) (1 – t^16)(1 – t^18)(1 – t^20)(1 – t^22)(1 – t^24)(1 – t^26) (1 – t^28)(1 – t^30)(1 – t^32)(1 – t^34)(1 – t^36)(1 – t^38) (1 – t^40)(1 – t^42)(1 – t^44)(1 – t^46)(1 – t^48)(1 – t^50) (1 – t^52)

### n=28

(1 + 3t^4 + 6t^5 + 33t^6 + 84t^7 + 313t^8 + 923t^9 + 2868t^10 + 8080t^11 + 22524t^12 + 58838t^13 + 149516t^14 + 363234t^15 + 855596t^16 + 1945416t^17 + 4298842t^18 + 9222323t^19 + 19283068t^20 + 39309803t^21 + 78317345t^22 + 152611883t^23 + 291354114t^24 + 545411729t^25 + 1002420427t^26 + 1810300347t^27 + 3215590520t^28 + 5622148879t^29 + 9683466583t^30 + 16441158673t^31 + 27536279585t^32 + 45519999373t^33 + 74315164667t^34 + 119882011799t^35 + 191183927201t^36 + 301555017639t^37 + 470642245817t^38 + 727108049685t^39 + 1112394475075t^40 + 1685875995495t^41 + 2531908995788t^42 + 3769333197670t^43 + 5564257777390t^44 + 8147043101948t^45 + 11834804680296t^46 + 17060888879441t^47 + 24413375813709t^48 + 34684848835745t^49 + 48936533949967t^50 + 68580171806079t^51 + 95482189631234t^52 + 132095090017633t^53 + 181622612316849t^54 + 248225670507478t^55 + 337278254449768t^56 + 455683017152425t^57 + 612259117712762t^58 + 818215484080944t^59 + 1087726286758771t^60 + 1438626004066524t^61 + 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### n=29

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### n=30

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68229136150124t^361 + 44821771504646t^362 + 29193590911219t^363 + 18846762588092t^364 + 12055881817138t^365 + 7638903663278t^366 + 4792678704235t^367 + 2976329567994t^368 + 1828793076008t^369 + 1111348046783t^370 + 667634756561t^371 + 396305908028t^372 + 232325396139t^373 + 134435033076t^374 + 76738319510t^375 + 43186187732t^376 + 23943942008t^377 + 13070339726t^378 + 7018374554t^379 + 3704634829t^380 + 1920138120t^381 + 976551702t^382 + 486624714t^383 + 237451142t^384 + 113218875t^385 + 52740933t^386 + 23922744t^387 + 10576403t^388 + 4530264t^389 + 1888017t^390 + 756280t^391 + 294960t^392 + 108918t^393 + 39570t^394 + 13130t^395 + 4479t^396 + 1267t^397 + 430t^398 + 88t^399 + 39t^400 + 2t^401 + 4t^402 + t^406) / (1 – t^2)(1 – t^4)(1 – t^5)(1 – t^6)^2(1 – t^7)(1 – t^8)(1 – t^9) (1 – t^10)(1 – t^11)(1 – t^12)(1 – t^13)(1 – t^14)(1 – t^15) (1 – t^16)(1 – t^17)(1 – t^18)(1 – t^19)(1 – t^20)(1 – t^21) (1 – t^22)(1 – t^23)(1 – t^24)(1 – t^25)(1 – t^26)(1 – t^27) (1 – t^28)(1 – t^29)