Invariants for one form, for several forms.

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

# The Poincaré series for two binary forms

Let Vn be the vector space (of dimension n+1) of binary forms of degree n. Let V be the direct sum of two of these. The present page gives the Poincaré series P(t) of the graded algebra of the SL2(C)-invariant polynomials on V (graded by degree). Computations are due to Leonid Bedratyuk and aeb.

## References

L. Bedratyuk, The Poincaré series of the algebras of simultaneous invariants and covariants of two binary forms, Linear and Multilinear Algebra 58 (2010) 789-803.

M. Brion, Invariants de plusieurs formes binaires, Bulletin de la S.M.F. 110 (1982) 429-445.

## Two forms

### V1+V1

1 / (1 – t2) =
1 + t2 + t4 + t6 + t8 + t10 + t12 + t14 + t16 + t18 + t20 + ...

Bivariate: 1 / (1 – tu)

### V2+V1

1 / (1 – t2)(1 – t3) =
1 + t2 + t3 + t4 + t5 + 2t6 + t7 + 2t8 + 2t9 + 2t10 + 2t11 + 3t12 + 2t13 + 3t14 + 3t15 + 3t16 + 3t17 + 4t18 + 3t19 + 4t20 + ...

Bivariate: 1 / (1 – u2)(1 – t2u)

### V2+V2

1 / (1 – t2)3 =
1 + 3t2 + 6t4 + 10t6 + 15t8 + 21t10 + 28t12 + 36t14 + 45t16 + 55t18 + 66t20 + ...

Bivariate: 1 / (1 – t2)(1 – tu)(1 – u2)

### V3+V1

(1 + t6) / (1 – t4)3 =
(1 – t2 + t4) / (1 – t2)(1 – t4)2 =
1 + 3t4 + t6 + 6t8 + 3t10 + 10t12 + 6t14 + 15t16 + 10t18 + 21t20 + ...

Bivariate: (1 + t3u3) / (1 – t3u)(1 – t2u2)(1 – u4) =
(1 – tu + t2u2) / (1 – tu)(1 – t3u)(1 – u4)

### V3+V2

(1 + t7) / (1 – t2)(1 – t3)(1 – t4)(1 – t5) =
(1 – t + t2 – t3 + t4 – t5 + t6) / (1 – t)(1 – t3)(1 – t4)(1 – t5) =
1 + t2 + t3 + 2t4 + 2t5 + 3t6 + 4t7 + 5t8 + 6t9 + 8t10 + 9t11 + 12t12 + 13t13 + 16t14 + 19t15 + 22t16 + 25t17 + 29t18 + 33t19 + 38t20 + ...

Bivariate: (1 + t3u4) / (1 – t2)(1 – tu2)(1 – u4)(1 – t3u2)

### V3+V3

(1 + t4 + t6 + t10) / (1 – t2)(1 – t4)4 =
(1 + t4)(1 – t2 + t4) / (1 – t2)2(1 – t4)3 =
1 + t2 + 6t4 + 7t6 + 21t8 + 26t10 + 56t12 + 70t14 + 125t16 + 155t18 + 246t20 + ...

Bivariate: (1 + t2u2)(1 + t3u3) / (1 – tu)(1 – t4)(1 – t3u)(1 – tu3)(1 – u4)

### V4+V1

(1 + t9) / (1 – t2)(1 – t3)(1 – t5)(1 – t6) =
(1 – t3 + t6) / (1 – t2)(1 – t3)2(1 – t5) =
1 + t2 + t3 + t4 + 2t5 + 3t6 + 2t7 + 4t8 + 5t9 + 5t10 + 7t11 + 9t12 + 8t13 + 12t14 + 14t15 + 14t16 + 18t17 + 21t18 + 21t19 + 27t20 + ...

Bivariate: (1 + t6u3) / (1 – u2)(1 – u3)(1 – t4u)(1 – t4u2) =
(1 – t2u + t4u2) / (1 – u2)(1 – t2u)(1 – u3)(1 – t4u)

### V4+V2

(1 + t6) / (1 – t2)2(1 – t3)2(1 – t4) =
(1 – t2 + t4) / (1 – t2)3(1 – t3)2 =
1 + 2t2 + 2t3 + 4t4 + 4t5 + 10t6 + 8t7 + 17t8 + 18t9 + 28t10 + 30t11 + 48t12 + 48t13 + 72t14 + 78t15 + 105t16 + 114t17 + 154t18 + 162t19 + 212t20 + ...

Bivariate: (1 + t3u3) / (1 – t2)(1 – u2)(1 – t2u)(1 – u3)(1 – t2u2) =
(1 – tu + t2u2) / (1 – t2)(1 – tu)(1 – u2)(1 – t2u)(1 – u3)

### V4+V3

(1 + t5 + t6 + 2t7 + 3t8 + 4t9 + 3t10 + 2t11 + t12 + t13 + t18) / (1 – t2)(1 – t3)(1 – t4)(1 – t5)(1 – t6)(1 – t7) =
(1 – t + t2 – 2t3 + 2t4 – t5 + 3t6 – t7 + 3t8 – t9 + 2t10 – 2t11 + t12 – t13 + t14) / (1 – t)(1 – t3)2(1 – t4)(1 – t5)(1 – t7) =
1 + t2 + t3 + 2t4 + 3t5 + 5t6 + 7t7 + 11t8 + 16t9 + 22t10 + 30t11 + 42t12 + 55t13 + 73t14 + 95t15 + 121t16 + 153t17 + 193t18 + 238t19 + 294t20 + ...

Bivariate: (1 – t2u + 2t4u2 + 2t4u3 + t6u2 + 2t4u4 + t6u3 + t4u5 – t8u3 – t6u5 – 2t8u4 – t6u6 – 2t8u5 – 2t8u6 + t10u7 – t12u8) / (1 – u2)(1 – t2u)(1 – u3)(1 – t4)(1 – t4u)(1 – t2u3)(1 – t4u3)

### V4+V4

(1 + t4 + t8) / (1 – t2)3(1 – t3)4 =
(1 – t + t2)(1 – t2 + t4) / (1 – t)(1 – t2)3(1 – t3)3 =
1 + 3t2 + 4t3 + 7t4 + 12t5 + 23t6 + 28t7 + 52t8 + 72t9 + 104t10 + 148t11 + 214t12 + 276t13 + 392t14 + 512t15 + 673t16 + 876t17 + 1141t18 + 1424t19 + 1831t20 + ...

Bivariate: (1 + t2u2 + t4u4) / (1 – t2)(1 – tu)(1 – u2)(1 – t3)(1 – t2u)(1 – tu2)(1 – u3)

### V5+V1

(1 + t6 + 6t8 + 3t10 + 6t12 + t14 + t20) / (1 – t4)2(1 – t6)2(1 – t8) =
1 + 2t4 + 3t6 + 10t8 + 9t10 + 29t12 + 31t14 + 61t16 + 75t18 + 125t20 + ...

Bivariate: (1 – tu – t3u + t2u2 + t4u2 + tu5 – u6 + t6u2 + t4u4 + t3u5 + 2tu7 – t7u3 – t5u5 – t4u6 + t3u7 – t2u8 + t5u7 – t4u8 + t3u9 + t2u10 + u12 – 2t6u8 – t4u10 – t3u11 – tu13 + t7u9 – t6u10 – t3u13 – t5u13 + t4u14 + t6u14 – t7u15) / (1 – tu)(1 – t3u)(1 – u4)(1 – t5u)(1 – u6)(1 – u8)

### V5+V2

(1 + t2 + t4 + t6 + 2t7 + 3t8 + 5t9 + 5t10 + 8t11 + 7t12 + 8t13 + 5t14 + 5t15 + 3t16 + 2t17 + t18 + t20 + t22 + t24) / (1 – t3)(1 – t4)(1 – t5)(1 – t6)(1 – t7)(1 – t8) =
1 + t2 + t3 + 2t4 + 2t5 + 4t6 + 6t7 + 9t8 + 12t9 + 17t10 + 24t11 + 33t12 + 42t13 + 56t14 + 73t15 + 94t16 + 117t17 + 148t18 + 183t19 + 226t20 + ...

Bivariate: (1 + t2u4 – u6 + t3u4 + tu6 + t4u4 + t2u6 + t5u4 + t3u6 + tu8 + t4u6 + t2u8 + t7u4 + t3u8 + tu10 + t6u6 + u12 + t5u8 – t3u10 – t8u6 – t2u12 – t7u8 – t5u10 – tu14 – t6u10 – t4u12 – t7u10 – t5u12 – t3u14 – t6u12 – t4u14 – t7u12 – t5u14 + t8u12 – t6u14 – t8u18) / (1 – t2)(1 – tu2)(1 – u4)(1 – t3u2)(1 – u6)(1 – t5u2)(1 – u8)

### V5+V3

(1 + 3t4 + 5t6 + 19t8 + 29t10 + 46t12 + 48t14 + 48t16 + 46t18 + 29t20 + 19t22 + 5t24 + 3t26 + t30) / (1 – t4)3(1 – t6)2(1 – t8)2 =
(1 – t2 + 4t4 + t6 + 18t8 + 11t10 + 35t12 + 13t14 + 35t16 + 11t18 + 18t20 + t22 + 4t24 – t26 + t28) / (1 – t2)(1 – t4)2(1 – t6)2(1 – t8)2 =
1 + 6t4 + 7t6 + 36t8 + 56t10 + 156t12 + 247t14 + 522t16 + 807t18 + 1443t20 + ...

(the form given by Brion has errors)

Bivariate: (1 – tu + 2t2u2 + t4u2 + t3u3 + t2u4 + tu5 – u6 + t6u2 + t5u3 + 3t4u4 + 2t3u5 + 2t2u6 + 2tu7 + t6u4 + 2t5u5 + 2t4u6 + 2t3u7 – t2u8 + 2tu9 – t8u4 + t6u6 + t5u7 + t4u8 + t3u9 – t2u10 + u12 – t10u4 – t9u5 – 2t8u6 – t7u7 + t5u9 – 2t4u10 – t2u12 – tu13 – t12u4 – t11u5 – 3t10u6 – 2t9u7 – t8u8 – t7u9 – 2t6u10 – t5u11 – 2t4u12 – 2t3u13 – tu15 + t13u5 – t10u8 – 2t9u9 – 2t6u12 – t5u13 + t2u16 – t14u6 – 2t12u8 – 2t11u9 – t10u10 – 2t9u11 – t8u12 – t7u13 – 2t6u14 – 3t5u15 – t4u16 – t3u17 – t14u8 – t13u9 – 2t11u11 + t10u12 – t8u14 – 2t7u15 – t6u16 – t5u17 + t15u9 – t13u11 + t12u12 + t11u13 + t10u14 + t9u15 – t7u17 + 2t14u12 – t13u13 + 2t12u14 + 2t11u15 + 2t10u16 + t9u17 + 2t14u14 + 2t13u15 + 2t12u16 + 3t11u17 + t10u18 + t9u19 – t15u15 + t14u16 + t13u17 + t12u18 + t11u19 + 2t13u19 – t14u20 + t15u21) / (1 – tu)(1 – t4)(1 – t3u)(1 – tu3)(1 – u4)(1 – t5u)(1 – u6)(1 – t5u3)(1 – u8)

### V5+V4

(1 + 2t5 + 4t6 + 6t7 + 9t8 + 15t9 + 18t10 + 23t11 + 29t12 + 31t13 + 33t14 + 37t15 + 37t16 + 37t17 + 37t18 + 33t19 + 31t20 + 29t21 + 23t22 + 18t23 + 15t24 + 9t25 + 6t26 + 4t27 + 2t28 + t33) / (1 – t2)(1 – t3)(1 – t4)(1 – t5)(1 – t6)(1 – t7)(1 – t8)(1 – t9) =
(1 – t + t2 – t3 + t4 + t5 + 3t6 + 3t7 + 6t8 + 9t9 + 9t10 + 14t11 + 15t12 + 16t13 + 17t14 + 20t15 + 17t16 + 20t17 + 17t18 + 16t19 + 15t20 + 14t21 + 9t22 + 9t23 + 6t24 + 3t25 + 3t26 + t27 + t28 – t29 + t30 – t31 + t32) / (1 – t)(1 – t3)(1 – t4)(1 – t5)(1 – t6)(1 – t7)(1 – t8)(1 – t9) =
1 + t2 + t3 + 2t4 + 4t5 + 8t6 + 12t7 + 22t8 + 37t9 + 56t10 + 88t11 + 135t12 + 195t13 + 284t14 + 405t15 + 562t16 + 775t17 + 1054t18 + 1405t19 + 1865t20 + ...

Bivariate: (1 – tu2 + tu4 + t4u2 + 4t2u4 – u6 + 4t3u4 + tu6 + 6t4u4 + 2t2u6 + 4t5u4 + 4t3u6 + 3tu8 + 4t6u4 + 4t4u6 + 3t2u8 + 2t7u4 + 3t5u6 + 4t3u8 + tu10 + t8u4 + 3t6u6 + 3t4u8 – t2u10 + u12 – t7u6 + 3t5u8 – 2t8u6 – 3t4u10 – 2t2u12 – 3t9u6 – 3t7u8 – 2t5u10 – 3t3u12 – tu14 – 2t10u6 – 4t8u8 – 6t6u10 – 6t4u12 – t2u14 – t11u6 – 4t9u8 – 5t7u10 – 6t5u12 – 3t3u14 – tu16 – 3t10u8 – 6t8u10 – 9t6u12 – 4t4u14 – t2u16 – 2t11u8 – 5t9u10 – 7t7u12 – 5t5u14 – 3t3u16 – t12u8 – 4t10u10 – 8t8u12 – 5t6u14 – 3t4u16 + t2u18 – 2t11u10 – 5t9u12 – 3t7u14 – 4t5u16 + t3u18 – t12u10 – 5t10u12 – 3t8u14 – 3t6u16 + 2t4u18 – 2t11u12 – t9u14 – 3t7u16 + t5u18 – 3t12u12 + t10u14 – t8u16 + 3t6u18 – t13u12 + 3t11u14 + t9u16 + 2t7u18 – 2t14u12 + 3t12u14 + 3t10u16 + 5t8u18 + t6u20 – t15u12 + 4t13u14 + 3t11u16 + 5t9u18 + 2t7u20 – t16u12 + 3t14u14 + 5t12u16 + 8t10u18 + 4t8u20 + t6u22 + 3t15u14 + 5t13u16 + 7t11u18 + 5t9u20 + 2t7u22 + t16u14 + 4t14u16 + 9t12u18 + 6t10u20 + 3t8u22 + t17u14 + 3t15u16 + 6t13u18 + 5t11u20 + 4t9u22 + t7u24 + t16u16 + 6t14u18 + 6t12u20 + 4t10u22 + 2t8u24 + t17u16 + 3t15u18 + 2t13u20 + 3t11u22 + 3t9u24 + 2t16u18 + 3t14u20 + 2t10u24 – 3t13u22 + t11u24 – t18u18 + t16u20 – 3t14u22 – 3t12u24 – t10u26 – t17u20 – 4t15u22 – 3t13u24 – 2t11u26 – 3t16u22 – 4t14u24 – 4t12u26 – 3t17u22 – 4t15u24 – 4t13u26 – 2t16u24 – 6t14u26 – t17u24 – 4t15u26 + t18u24 – 4t16u26 – t14u28 – t17u26 + t17u28 – t18u30) / (1 – t2)(1 – t3)(1 – tu2)(1 – u4)(1 – t3u2)(1 – tu4)(1 – u6)(1 – t5u2)(1 – t3u4)(1 – u8)(1 – t5u4)

### V5+V5

(1 + 4t4 + 4t6 + 44t8 + 70t10 + 184t12 + 215t14 + 382t16 + 391t18 + 508t20 + 391t22 + 382t24 + 215t26 + 184t28 + 70t30 + 44t32 + 4t34 + 4t36 + t40) / (1 – t2)(1 – t4)2(1 – t6)3(1 – t8)3 =
1 + t2 + 7t4 + 14t6 + 72t8 + 168t10 + 492t12 + 1045t14 + 2389t16 + 4622t18 + 9033t20 + ...

(the form given by Brion has errors)

Bivariate: (1 + 2t2u2 – t6 + t4u2 + 3t3u3 + t2u4 – u6 + 2t7u + 5t6u2 + 6t5u3 + 10t4u4 + 6t3u5 + 5t2u6 + 2tu7 + 2t9u + 2t8u2 + 7t7u3 + 11t6u4 + 16t5u5 + 11t4u6 + 7t3u7 + 2t2u8 + 2tu9 + t12 + 2t11u + 2t10u2 + 5t9u3 + 11t8u4 + 15t7u5 + 22t6u6 + 15t5u7 + 11t4u8 + 5t3u9 + 2t2u10 + 2tu11 + u12 – t13u – 3t12u2 – 2t11u3 – t10u4 + 7t9u5 + 11t8u6 + 17t7u7 + 11t6u8 + 7t5u9 – t4u10 – 2t3u11 – 3t2u12 – tu13 – t15u – 3t14u2 – 5t13u3 – 9t12u4 – 9t11u5 – 3t10u6 + 3t9u7 + 13t8u8 + 3t7u9 – 3t6u10 – 9t5u11 – 9t4u12 – 5t3u13 – 3t2u14 – tu15 – t17u – t16u2 – 4t15u3 – 9t14u4 – 14t13u5 – 17t12u6 – 11t11u7 – 7t10u8 – t9u9 – 7t8u10 – 11t7u11 – 17t6u12 – 14t5u13 – 9t4u14 – 4t3u15 – t2u16 – tu17 – 2t17u3 – 8t16u4 – 15t15u5 – 24t14u6 – 26t13u7 – 23t12u8 – 20t11u9 – 11t10u10 – 20t9u11 – 23t8u12 – 26t7u13 – 24t6u14 – 15t5u15 – 8t4u16 – 2t3u17 + t20u2 – 7t17u5 – 15t16u6 – 27t15u7 – 33t14u8 – 29t13u9 – 21t12u10 – 14t11u11 – 21t10u12 – 29t9u13 – 33t8u14 – 27t7u15 – 15t6u16 – 7t5u17 + t2u20 – 7t18u6 – 15t17u7 – 28t16u8 – 31t15u9 – 28t14u10 – 23t13u11 – 15t12u12 – 23t11u13 – 28t10u14 – 31t9u15 – 28t8u16 – 15t7u17 – 7t6u18 + 2t22u4 + 3t21u5 + 6t20u6 – 4t18u8 – 13t17u9 – 12t16u10 – 7t15u11 + 3t14u12 + 9t13u13 + 3t12u14 – 7t11u15 – 12t10u16 – 13t9u17 – 4t8u18 + 6t6u20 + 3t5u21 + 2t4u22 + 2t23u5 + 5t22u6 + 11t21u7 + 10t20u8 + 10t19u9 + 3t18u10 + 9t17u11 + 20t16u12 + 35t15u13 + 45t14u14 + 35t13u15 + 20t12u16 + 9t11u17 + 3t10u18 + 10t9u19 + 10t8u20 + 11t7u21 + 5t6u22 + 2t5u23 + 3t24u6 + 7t23u7 + 16t22u8 + 17t21u9 + 22t20u10 + 21t19u11 + 32t18u12 + 45t17u13 + 61t16u14 + 69t15u15 + 61t14u16 + 45t13u17 + 32t12u18 + 21t11u19 + 22t10u20 + 17t9u21 + 16t8u22 + 7t7u23 + 3t6u24 + 3t25u7 + 7t24u8 + 16t23u9 + 17t22u10 + 22t21u11 + 21t20u12 + 32t19u13 + 45t18u14 + 61t17u15 + 69t16u16 + 61t15u17 + 45t14u18 + 32t13u19 + 21t12u20 + 22t11u21 + 17t10u22 + 16t9u23 + 7t8u24 + 3t7u25 + 2t26u8 + 5t25u9 + 11t24u10 + 10t23u11 + 10t22u12 + 3t21u13 + 9t20u14 + 20t19u15 + 35t18u16 + 45t17u17 + 35t16u18 + 20t15u19 + 9t14u20 + 3t13u21 + 10t12u22 + 10t11u23 + 11t10u24 + 5t9u25 + 2t8u26 + 2t27u9 + 3t26u10 + 6t25u11 – 4t23u13 – 13t22u14 – 12t21u15 – 7t20u16 + 3t19u17 + 9t18u18 + 3t17u19 – 7t16u20 – 12t15u21 – 13t14u22 – 4t13u23 + 6t11u25 + 3t10u26 + 2t9u27 – 7t25u13 – 15t24u14 – 28t23u15 – 31t22u16 – 28t21u17 – 23t20u18 – 15t19u19 – 23t18u20 – 28t17u21 – 31t16u22 – 28t15u23 – 15t14u24 – 7t13u25 + t29u11 – 7t26u14 – 15t25u15 – 27t24u16 – 33t23u17 – 29t22u18 – 21t21u19 – 14t20u20 – 21t19u21 – 29t18u22 – 33t17u23 – 27t16u24 – 15t15u25 – 7t14u26 + t11u29 – 2t28u14 – 8t27u15 – 15t26u16 – 24t25u17 – 26t24u18 – 23t23u19 – 20t22u20 – 11t21u21 – 20t20u22 – 23t19u23 – 26t18u24 – 24t17u25 – 15t16u26 – 8t15u27 – 2t14u28 – t30u14 – t29u15 – 4t28u16 – 9t27u17 – 14t26u18 – 17t25u19 – 11t24u20 – 7t23u21 – t22u22 – 7t21u23 – 11t20u24 – 17t19u25 – 14t18u26 – 9t17u27 – 4t16u28 – t15u29 – t14u30 – t30u16 – 3t29u17 – 5t28u18 – 9t27u19 – 9t26u20 – 3t25u21 + 3t24u22 + 13t23u23 + 3t22u24 – 3t21u25 – 9t20u26 – 9t19u27 – 5t18u28 – 3t17u29 – t16u30 – t30u18 – 3t29u19 – 2t28u20 – t27u21 + 7t26u22 + 11t25u23 + 17t24u24 + 11t23u25 + 7t22u26 – t21u27 – 2t20u28 – 3t19u29 – t18u30 + t31u19 + 2t30u20 + 2t29u21 + 5t28u22 + 11t27u23 + 15t26u24 + 22t25u25 + 15t24u26 + 11t23u27 + 5t22u28 + 2t21u29 + 2t20u30 + t19u31 + 2t30u22 + 2t29u23 + 7t28u24 + 11t27u25 + 16t26u26 + 11t25u27 + 7t24u28 + 2t23u29 + 2t22u30 + 2t30u24 + 5t29u25 + 6t28u26 + 10t27u27 + 6t26u28 + 5t25u29 + 2t24u30 – t31u25 + t29u27 + 3t28u28 + t27u29 – t25u31 + 2t29u29 + t31u31) / (1 – tu)(1 – t4)(1 – t3u)(1 – tu3)(1 – u4)(1 – t6)(1 – t5u)(1 – tu5)(1 – u6)(1 – t8)(1 – t5u3)(1 – t3u5)(1 – u8)

### V6+V1

(1 + t2 + t4 + t6 + t7 + 2t8 + 4t9 + 5t10 + 6t11 + 8t12 + 7t13 + 9t14 + 9t15 + 7t16 + 8t17 + 6t18 + 5t19 + 4t20 + 2t21 + t22 + t23 + t25 + t27 + t29) / (1 – t4)(1 – t5)(1 – t6)2(1 – t7)(1 – t10) =
(1 + 2t + t2 – 2t3 – 3t4 – t5 + 2t6 + 4t7 + 5t8 + 5t9 + 5t10 + 4t11 + 2t12 – t13 – 3t14 – 2t15 + t16 + 2t17 + t18) / (1 + t)2(1 – t2)(1 – t3)2(1 – t5)2(1 – t7) =
1 + t2 + 2t4 + t5 + 4t6 + 3t7 + 6t8 + 7t9 + 13t10 + 13t11 + 23t12 + 23t13 + 38t14 + 43t15 + 61t16 + 69t17 + 94t18 + 108t19 + 143t20 + ...

### V6+V2

(1 + 2t4 + 3t6 + 3t7 + 4t8 + 7t9 + 4t10 + 7t11 + 6t12 + 7t13 + 4t14 + 7t15 + 4t16 + 3t17 + 3t18 + 2t20 + t24) / (1 – t2)2(1 – t4)2(1 – t6)2(1 – t10) =
(1 + t – 2t3 + t5 + 5t6 + 5t7 + 5t8 + t9 – 2t11 + t13 + t14) / (1 + t)(1 – t2)2(1 – t3)2(1 – t4)2(1 – t5) =
1 + 2t2 + 7t4 + 17t6 + 3t7 + 38t8 + 13t9 + 78t10 + 36t11 + 151t12 + 86t13 + 271t14 + 180t15 + 469t16 + 341t17 + 774t18 + 607t19 + 1233t20 + ...

### V6+V3

(1 + t5 + 3t6 + 5t7 + 7t8 + 9t9 + 10t10 + 10t11 + 10t12 + 10t13 + 10t14 + 10t15 + 9t16 + 7t17 + 5t18 + 3t19 + t20 + t25) / (1 – t2)(1 – t3)(1 – t4)2(1 – t5)2(1 – t6)(1 – t7) =
(1 – t2 – t3 + t4 + 2t5 + 3t6 + 3t7 + 3t8 + 2t9 + 2t10 + 2t11 + 3t12 + 3t13 + 3t14 + 2t15 + t16 – t17 – t18 + t20) / (1 – t2)2(1 – t3)2(1 – t4)(1 – t5)2(1 – t7) =
1 + t2 + t3 + 3t4 + 4t5 + 8t6 + 12t7 + 21t8 + 32t9 + 52t10 + 76t11 + 117t12 + 167t13 + 244t14 + 341t15 + 477t16 + 649t17 + 883t18 + 1175t19 + 1558t20 + ...

### V6+V4

(1 + 2t4 + 2t5 + 8t6 + 9t7 + 12t8 + 12t9 + 17t10 + 13t11 + 17t12 + 12t13 + 12t14 + 9t15 + 8t16 + 2t17 + 2t18 + t22) / (1 – t2)2(1 – t3)2(1 – t4)2(1 – t5)2(1 – t6) =
(1 + t – t3 + t4 + 4t5 + 11t6 + 16t7 + 17t8 + 13t9 + 13t10 + 13t11 + 17t12 + 16t13 + 11t14 + 4t15 + t16 – t17 + t19 + t20) / (1 + t)(1 – t2)2(1 – t3)3(1 – t4)2(1 – t5)2 =
1 + 2t2 + 2t3 + 7t4 + 8t5 + 24t6 + 31t7 + 68t8 + 100t9 + 186t10 + 270t11 + 462t12 + 660t13 + 1046t14 + 1482t15 + 2217t16 + 3069t17 + 4422t18 + 5996t19 + 8349t20 + ...

### V6+V5

(1 + t4 + 2t5 + 7t6 + 14t7 + 23t8 + 38t9 + 60t10 + 87t11 + 124t12 + 170t13 + 223t14 + 285t15 + 355t16 + 426t17 + 499t18 + 571t19 + 633t20 + 684t21 + 721t22 + 739t23 + 739t24 + 721t25 + 684t26 + 633t27 + 571t28 + 499t29 + 426t30 + 355t31 + 285t32 + 223t33 + 170t34 + 124t35 + 87t36 + 60t37 + 38t38 + 23t39 + 14t40 + 7t41 + 2t42 + t43 + t47) / (1 – t2)(1 – t3)(1 – t4)(1 – t5)2(1 – t6)(1 – t7)(1 – t8)(1 – t9)(1 – t11) =
(1 – t + t2 – t3 + 2t4 + 7t6 + 7t7 + 16t8 + 22t9 + 38t10 + 49t11 + 75t12 + 95t13 + 128t14 + 157t15 + 198t16 + 228t17 + 271t18 + 300t19 + 333t20 + 351t21 + 370t22 + 369t23 + 370t24 + 351t25 + 333t26 + 300t27 + 271t28 + 228t29 + 198t30 + 157t31 + 128t32 + 95t33 + 75t34 + 49t35 + 38t36 + 22t37 + 16t38 + 7t39 + 7t40 + 2t42 – t43 + t44 – t45 + t46) / (1 – t)(1 – t3)(1 – t4)(1 – t5)2(1 – t6)(1 – t7)(1 – t8)(1 – t9)(1 – t11) =
1 + t2 + t3 + 3t4 + 5t5 + 12t6 + 22t7 + 42t8 + 76t9 + 135t10 + 228t11 + 380t12 + 609t13 + 960t14 + 1478t15 + 2233t16 + 3307t17 + 4828t18 + 6931t19 + 9820t20 + ...

### V6+V6

(1 + t2 + 4t4 + 3t5 + 17t6 + 25t7 + 53t8 + 72t9 + 135t10 + 163t11 + 240t12 + 295t13 + 365t14 + 415t15 + 463t16 + 482t17 + 482t18 + 463t19 + 415t20 + 365t21 + 295t22 + 240t23 + 163t24 + 135t25 + 72t26 + 53t27 + 25t28 + 17t29 + 3t30 + 4t31 + t33 + t35) / (1 – t2)2(1 – t4)3(1 – t5)3(1 – t6)3 =
(1 + 2t + 2t2 – t3 – t4 + 5t5 + 27t6 + 59t7 + 99t8 + 124t9 + 156t10 + 191t11 + 249t12 + 294t13 + 320t14 + 294t15 + 249t16 + 191t17 + 156t18 + 124t19 + 99t20 + 59t21 + 27t22 + 5t23 – t24 – t25 + 2t26 + 2t27 + t28) / (1 + t)2(1 – t2)2(1 – t3)3(1 – t4)3(1 – t5)3 =
1 + 3t2 + 12t4 + 6t5 + 44t6 + 40t7 + 150t8 + 176t9 + 484t10 + 628t11 + 1398t12 + 1906t13 + 3698t14 + 5148t15 + 9051t16 + 12608t17 + 20661t18 + 28600t19 + 44422t20 + ...

### V7+V1

(1 + t4 + 15t8 + 13t10 + 49t12 + 49t14 + 92t16 + 89t18 + 133t20 + 99t22 + 133t24 + 89t26 + 92t28 + 49t30 + 49t32 + 13t34 + 15t36 + t40 + t44) / (1 – t4)(1 – t6)2(1 – t8)2(1 – t10)(1 – t12) =
1 + 2t4 + 2t6 + 19t8 + 18t10 + 74t12 + 102t14 + 231t16 + 342t18 + 647t20 + ...

### V7+V2

(1 + t2 + t3 + t4 + t5 + 3t6 + 5t7 + 9t8 + 15t9 + 21t10 + 36t11 + 51t12 + 66t13 + 90t14 + 118t15 + 140t16 + 171t17 + 203t18 + 227t19 + 259t20 + 278t21 + 298t22 + 315t23 + 327t24 + 320t25 + 327t26 + 315t27 + 298t28 + 278t29 + 259t30 + 227t31 + 203t32 + 171t33 + 140t34 + 118t35 + 90t36 + 66t37 + 51t38 + 36t39 + 21t40 + 15t41 + 9t42 + 5t43 + 3t44 + t45 + t46 + t47 + t48 + t50) / (1 – t4)(1 – t5)(1 – t6)(1 – t7)(1 – t8)(1 – t9)(1 – t10)(1 – t12) =
1 + t2 + t3 + 2t4 + 2t5 + 5t6 + 8t7 + 14t8 + 21t9 + 32t10 + 52t11 + 81t12 + 111t13 + 164t14 + 235t15 + 326t16 + 444t17 + 607t18 + 808t19 + 1076t20 + ...

### V7+V3

(1 + 3t4 + 7t6 + 42t8 + 90t10 + 236t12 + 415t14 + 768t16 + 1147t18 + 1713t20 + 2176t22 + 2732t24 + 2976t26 + 3170t28 + 2976t30 + 2732t32 + 2176t34 + 1713t36 + 1147t38 + 768t40 + 415t42 + 236t44 + 90t46 + 42t48 + 7t50 + 3t52 + t56) / (1 – t4)2(1 – t6)2(1 – t8)2(1 – t10)2(1 – t12) =
1 + 5t4 + 9t6 + 53t8 + 116t10 + 361t12 + 746t14 + 1734t16 + 3309t18 + 6530t20 + ...

### V7+V4

(1 + t2 + t4 + 3t5 + 7t6 + 13t7 + 28t8 + 50t9 + 80t10 + 132t11 + 202t12 + 297t13 + 428t14 + 597t15 + 804t16 + 1071t17 + 1379t18 + 1734t19 + 2150t20 + 2601t21 + 3078t22 + 3587t23 + 4098t24 + 4584t25 + 5055t26 + 5461t27 + 5798t28 + 6060t29 + 6222t30 + 6262t31 + 6222t32 + 6060t33 + 5798t34 + 5461t35 + 5055t36 + 4584t37 + 4098t38 + 3587t39 + 3078t40 + 2601t41 + 2150t42 + 1734t43 + 1379t44 + 1071t45 + 804t46 + 597t47 + 428t48 + 297t49 + 202t50 + 132t51 + 80t52 + 50t53 + 28t54 + 13t55 + 7t56 + 3t57 + t58 + t60 + t62) / (1 – t3)(1 – t4)(1 – t5)(1 – t6)(1 – t7)(1 – t8)(1 – t9)(1 – t10)(1 – t11)(1 – t12) =
1 + t2 + t3 + 2t4 + 5t5 + 10t6 + 17t7 + 37t8 + 67t9 + 113t10 + 198t11 + 330t12 + 523t13 + 831t14 + 1280t15 + 1924t16 + 2865t17 + 4177t18 + 5982t19 + 8496t20 + ...

### V7+V5

(1 – t2 + 6t4 + 10t6 + 96t8 + 197t10 + 733t12 + 1430t14 + 3441t16 + 6016t18 + 11245t20 + 17165t22 + 26931t24 + 36177t26 + 48945t28 + 58453t30 + 69487t32 + 73849t34 + 78024t36 + 73849t38 + 69487t40 + 58453t42 + 48945t44 + 36177t46 + 26931t48 + 17165t50 + 11245t52 + 6016t54 + 3441t56 + 1430t58 + 733t60 + 197t62 + 96t64 + 10t66 + 6t68 – t70 + t72) / (1 – t2)(1 – t4)(1 – t6)2(1 – t8)3(1 – t10)2(1 – t12)2 =
1 + 7t4 + 18t6 + 122t8 + 341t10 + 1219t12 + 3103t14 + 8158t16 + 18397t18 + 40754t20 + ...

### V7+V6

(1 – t + t2 – t3 + 2t4 + 2t5 + 9t6 + 16t7 + 33t8 + 60t9 + 104t10 + 178t11 + 289t12 + 452t13 + 690t14 + 1024t15 + 1468t16 + 2065t17 + 2835t18 + 3802t19 + 5001t20 + 6445t21 + 8144t22 + 10118t23 + 12346t24 + 14805t25 + 17480t26 + 20310t27 + 23226t28 + 26179t29 + 29071t30 + 31807t31 + 34325t32 + 36514t33 + 38305t34 + 39644t35 + 40466t36 + 40732t37 + 40466t38 + 39644t39 + 38305t40 + 36514t41 + 34325t42 + 31807t43 + 29071t44 + 26179t45 + 23226t46 + 20310t47 + 17480t48 + 14805t49 + 12346t50 + 10118t51 + 8144t52 + 6445t53 + 5001t54 + 3802t55 + 2835t56 + 2065t57 + 1468t58 + 1024t59 + 690t60 + 452t61 + 289t62 + 178t63 + 104t64 + 60t65 + 33t66 + 16t67 + 9t68 + 2t69 + 2t70 – t71 + t72 – t73 + t74) / (1 – t)(1 – t3)(1 – t4)(1 – t5)(1 – t6)(1 – t7)(1 – t8)(1 – t9)(1 – t10)(1 – t11)(1 – t12)(1 – t13) =
1 + t2 + t3 + 3t4 + 6t5 + 16t6 + 34t7 + 73t8 + 147t9 + 284t10 + 530t11 + 960t12 + 1675t13 + 2855t14 + 4747t15 + 7706t16 + 12257t17 + 19141t18 + 29353t19 + 44306t20 + ...

### V7+V7

(1 + 6t4 + 20t6 + 172t8 + 520t10 + 2059t12 + 5418t14 + 14587t16 + 32583t18 + 70453t20 + 134803t22 + 247902t24 + 416220t26 + 669438t28 + 1001770t30 + 1436870t32 + 1936018t34 + 2505422t36 + 3064172t38 + 3602433t40 + 4018051t42 + 4311209t44 + 4394462t46 + 4311209t48 + 4018051t50 + 3602433t52 + 3064172t54 + 2505422t56 + 1936018t58 + 1436870t60 + 1001770t62 + 669438t64 + 416220t66 + 247902t68 + 134803t70 + 70453t72 + 32583t74 + 14587t76 + 5418t78 + 2059t80 + 520t82 + 172t84 + 20t86 + 6t88 + t92) / (1 – t2)(1 – t4)(1 – t6)2(1 – t8)3(1 – t10)3(1 – t12)3 =
1 + t2 + 8t4 + 30t6 + 212t8 + 769t10 + 3074t12 + 9477t14 + 28033t16 + 73388t18 + 182218t20 + ...

### V8+V1

(1 + 2t + t2 – t3 – t4 + t6 + 3t7 + 8t8 + 15t9 + 20t10 + 21t11 + 22t12 + 24t13 + 26t14 + 30t15 + 33t16 + 30t17 + 26t18 + 24t19 + 22t20 + 21t21 + 20t22 + 15t23 + 8t24 + 3t25 + t26 – t28 – t29 + t30 + 2t31 + t32) / (1 + t)2(1 – t2)(1 – t3)2(1 – t5)2(1 – t7)2(1 – t9) =
1 + t2 + t3 + 2t4 + 2t5 + 5t6 + 6t7 + 11t8 + 17t9 + 25t10 + 36t11 + 55t12 + 76t13 + 112t14 + 156t15 + 213t16 + 290t17 + 395t18 + 517t19 + 688t20 + ...

### V8+V2

(1 + 2t + 2t2 – 2t4 – t5 + 7t6 + 19t7 + 32t8 + 40t9 + 46t10 + 46t11 + 46t12 + 45t13 + 46t14 + 46t15 + 46t16 + 40t17 + 32t18 + 19t19 + 7t20 – t21 – 2t22 + 2t24 + 2t25 + t26) / (1 + t)2(1 – t2)(1 – t3)3(1 – t4)2(1 – t5)2(1 – t7) =
1 + 2t2 + t3 + 5t4 + 5t5 + 15t6 + 17t7 + 41t8 + 54t9 + 108t10 + 144t11 + 261t12 + 354t13 + 581t14 + 795t15 + 1218t16 + 1648t17 + 2414t18 + 3217t19 + 4539t20 + ...

### V8+V3

(1 + t – t3 + 2t5 + 6t6 + 12t7 + 22t8 + 35t9 + 52t10 + 73t11 + 102t12 + 138t13 + 178t14 + 221t15 + 266t16 + 310t17 + 353t18 + 390t19 + 418t20 + 437t21 + 444t22 + 437t23 + 418t24 + 390t25 + 353t26 + 310t27 + 266t28 + 221t29 + 178t30 + 138t31 + 102t32 + 73t33 + 52t34 + 35t35 + 22t36 + 12t37 + 6t38 + 2t39 – t41 + t43 + t44) / (1 + t)(1 – t2)(1 – t3)2(1 – t4)(1 – t5)2(1 – t7)2(1 – t9)(1 – t11) =
1 + t2 + t3 + 3t4 + 4t5 + 9t6 + 16t7 + 30t8 + 53t9 + 92t10 + 154t11 + 255t12 + 407t13 + 639t14 + 981t15 + 1477t16 + 2185t17 + 3186t18 + 4567t19 + 6465t20 + ...

### V8+V4

(1 + t + 3t4 + 9t5 + 17t6 + 22t7 + 28t8 + 41t9 + 63t10 + 85t11 + 107t12 + 118t13 + 121t14 + 118t15 + 107t16 + 85t17 + 63t18 + 41t19 + 28t20 + 22t21 + 17t22 + 9t23 + 3t24 + t27 + t28) / (1 + t)(1 – t2)2(1 – t3)4(1 – t4)2(1 – t5)2(1 – t7) =
1 + 2t2 + 4t3 + 8t4 + 16t5 + 35t6 + 60t7 + 120t8 + 214t9 + 379t10 + 647t11 + 1105t12 + 1790t13 + 2905t14 + 4570t15 + 7082t16 + 10763t17 + 16163t18 + 23784t19 + 34662t20 + ...

### V8+V5

(1 + t2 – t3 + 2t4 + 2t5 + 12t6 + 22t7 + 52t8 + 94t9 + 175t10 + 292t11 + 485t12 + 757t13 + 1161t14 + 1707t15 + 2452t16 + 3409t17 + 4631t18 + 6117t19 + 7902t20 + 9965t21 + 12301t22 + 14862t23 + 17606t24 + 20446t25 + 23304t26 + 26077t27 + 28659t28 + 30948t29 + 32845t30 + 34265t31 + 35143t32 + 35443t33 + 35143t34 + 34265t35 + 32845t36 + 30948t37 + 28659t38 + 26077t39 + 23304t40 + 20446t41 + 17606t42 + 14862t43 + 12301t44 + 9965t45 + 7902t46 + 6117t47 + 4631t48 + 3409t49 + 2452t50 + 1707t51 + 1161t52 + 757t53 + 485t54 + 292t55 + 175t56 + 94t57 + 52t58 + 22t59 + 12t60 + 2t61 + 2t62 – t63 + t64 + t66) / (1 – t3)2(1 – t4)(1 – t5)2(1 – t6)(1 – t7)2(1 – t8)(1 – t9)(1 – t11)(1 – t13) =
1 + t2 + t3 + 3t4 + 6t5 + 15t6 + 31t7 + 66t8 + 133t9 + 256t10 + 476t11 + 860t12 + 1501t13 + 2556t14 + 4245t15 + 6889t16 + 10956t17 + 17101t18 + 26219t19 + 39567t20 + ...

### V8+V6

(1 + 3t + 4t2 + 2t3 + 2t4 + 15t5 + 59t6 + 150t7 + 312t8 + 578t9 + 1011t10 + 1673t11 + 2631t12 + 3917t13 + 5541t14 + 7450t15 + 9551t16 + 11651t17 + 13543t18 + 15011t19 + 15933t20 + 16238t21 + 15933t22 + 15011t23 + 13543t24 + 11651t25 + 9551t26 + 7450t27 + 5541t28 + 3917t29 + 2631t30 + 1673t31 + 1011t32 + 578t33 + 312t34 + 150t35 + 59t36 + 15t37 + 2t38 + 2t39 + 4t40 + 3t41 + t42) / (1 + t)3(1 – t2)(1 – t3)4(1 – t4)3(1 – t5)3(1 – t7)2 =
1 + 2t2 + 2t3 + 10t4 + 14t5 + 46t6 + 82t7 + 206t8 + 386t9 + 830t10 + 1512t11 + 2942t12 + 5170t13 + 9310t14 + 15734t15 + 26735t16 + 43482t17 + 70576t18 + 110918t19 + 173288t20 + ...

### V8+V7

(1 – t + t2 – t3 + 2t4 + 4t5 + 13t6 + 26t7 + 63t8 + 127t9 + 244t10 + 461t11 + 827t12 + 1422t13 + 2394t14 + 3883t15 + 6116t16 + 9429t17 + 14135t18 + 20720t19 + 29772t20 + 41868t21 + 57763t22 + 78324t23 + 104307t24 + 136647t25 + 176276t26 + 223892t27 + 280257t28 + 346011t29 + 421291t30 + 506262t31 + 600737t32 + 703875t33 + 814808t34 + 932203t35 + 1054039t36 + 1178359t37 + 1302789t38 + 1424415t39 + 1540648t40 + 1648689t41 + 1745522t42 + 1828778t43 + 1896233t44 + 1945686t45 + 1975994t46 + 1986284t47 + 1975994t48 + 1945686t49 + 1896233t50 + 1828778t51 + 1745522t52 + 1648689t53 + 1540648t54 + 1424415t55 + 1302789t56 + 1178359t57 + 1054039t58 + 932203t59 + 814808t60 + 703875t61 + 600737t62 + 506262t63 + 421291t64 + 346011t65 + 280257t66 + 223892t67 + 176276t68 + 136647t69 + 104307t70 + 78324t71 + 57763t72 + 41868t73 + 29772t74 + 20720t75 + 14135t76 + 9429t77 + 6116t78 + 3883t79 + 2394t80 + 1422t81 + 827t82 + 461t83 + 244t84 + 127t85 + 63t86 + 26t87 + 13t88 + 4t89 + 2t90 – t91 + t92 – t93 + t94) / (1 – t)(1 – t3)(1 – t4)(1 – t5)(1 – t6)(1 – t7)2(1 – t8)(1 – t9)(1 – t10)(1 – t11)(1 – t12)(1 – t13)(1 – t15) =
1 + t2 + t3 + 3t4 + 8t5 + 22t6 + 51t7 + 121t8 + 269t9 + 562t10 + 1141t11 + 2233t12 + 4203t13 + 7706t14 + 13739t15 + 23860t16 + 40540t17 + 67432t18 + 109961t19 + 176177t20 + ...

### V8+V8

(1 + 3t + 5t2 + 6t3 + 10t4 + 29t5 + 89t6 + 234t7 + 541t8 + 1133t9 + 2205t10 + 4039t11 + 7045t12 + 11752t13 + 18757t14 + 28568t15 + 41446t16 + 57271t17 + 75606t18 + 95718t19 + 116642t20 + 137119t21 + 155631t22 + 170483t23 + 180140t24 + 183488t25 + 180140t26 + 170483t27 + 155631t28 + 137119t29 + 116642t30 + 95718t31 + 75606t32 + 57271t33 + 41446t34 + 28568t35 + 18757t36 + 11752t37 + 7045t38 + 4039t39 + 2205t40 + 1133t41 + 541t42 + 234t43 + 89t44 + 29t45 + 10t46 + 6t47 + 5t48 + 3t49 + t50) / (1 + t)3(1 – t2)(1 – t3)5(1 – t4)3(1 – t5)3(1 – t7)3 =
1 + 3t2 + 4t3 + 13t4 + 26t5 + 71t6 + 156t7 + 369t8 + 796t9 + 1715t10 + 3496t11 + 7019t12 + 13514t13 + 25483t14 + 46576t15 + 83318t16 + 145364t17 + 248726t18 + 416694t19 + 686074t20 + ...

### V9+V1

(1 + t2 + 3t4 + 3t6 + 24t8 + 63t10 + 196t12 + 407t14 + 879t16 + 1580t18 + 2805t20 + 4402t22 + 6764t24 + 9499t26 + 12973t28 + 16465t30 + 20309t32 + 23499t34 + 26426t36 + 27986t38 + 28803t40 + 27986t42 + 26426t44 + 23499t46 + 20309t48 + 16465t50 + 12973t52 + 9499t54 + 6764t56 + 4402t58 + 2805t60 + 1580t62 + 879t64 + 407t66 + 196t68 + 63t70 + 24t72 + 3t74 + 3t76 + t78 + t80) / (1 + t2)(1 – t6)2(1 – t8)2(1 – t10)2(1 – t12)(1 – t14)(1 – t16) =
1 + 3t4 + 2t6 + 26t8 + 47t10 + 167t12 + 309t14 + 775t16 + 1416t18 + 2879t20 + ...

### V9+V2

(1 + t + 2t2 + 3t3 + 5t4 + 6t5 + 10t6 + 18t7 + 34t8 + 61t9 + 110t10 + 194t11 + 330t12 + 532t13 + 838t14 + 1284t15 + 1918t16 + 2774t17 + 3944t18 + 5474t19 + 7467t20 + 9957t21 + 13086t22 + 16879t23 + 21482t24 + 26852t25 + 33154t26 + 40309t27 + 48438t28 + 57336t29 + 67143t30 + 77569t31 + 88680t32 + 100060t33 + 111775t34 + 123345t35 + 134797t36 + 145545t37 + 155667t38 + 164605t39 + 172440t40 + 178587t41 + 183267t42 + 185996t43 + 187043t44 + 185996t45 + 183267t46 + 178587t47 + 172440t48 + 164605t49 + 155667t50 + 145545t51 + 134797t52 + 123345t53 + 111775t54 + 100060t55 + 88680t56 + 77569t57 + 67143t58 + 57336t59 + 48438t60 + 40309t61 + 33154t62 + 26852t63 + 21482t64 + 16879t65 + 13086t66 + 9957t67 + 7467t68 + 5474t69 + 3944t70 + 2774t71 + 1918t72 + 1284t73 + 838t74 + 532t75 + 330t76 + 194t77 + 110t78 + 61t79 + 34t80 + 18t81 + 10t82 + 6t83 + 5t84 + 3t85 + 2t86 + t87 + t88) / (1 + t)(1 + t2)(1 – t5)(1 – t6)(1 – t7)(1 – t8)(1 – t9)(1 – t10)(1 – t11)(1 – t12)(1 – t14)(1 – t16) =
1 + t2 + t3 + 3t4 + 2t5 + 6t6 + 11t7 + 22t8 + 34t9 + 62t10 + 106t11 + 179t12 + 273t13 + 437t14 + 671t15 + 1025t16 + 1486t17 + 2192t18 + 3128t19 + 4463t20 + ...

### V9+V3

(1 + 5t4 + 9t6 + 70t8 + 183t10 + 587t12 + 1331t14 + 3096t16 + 5994t18 + 11299t20 + 18936t22 + 30620t24 + 45397t26 + 64871t28 + 86472t30 + 111214t32 + 134703t34 + 157619t36 + 174621t38 + 187086t40 + 190233t42 + 187086t44 + 174621t46 + 157619t48 + 134703t50 + 111214t52 + 86472t54 + 64871t56 + 45397t58 + 30620t60 + 18936t62 + 11299t64 + 5994t66 + 3096t68 + 1331t70 + 587t72 + 183t74 + 70t76 + 9t78 + 5t80 + t84) / (1 – t4)2(1 – t6)2(1 – t8)2(1 – t10)2(1 – t12)(1 – t14)(1 – t16) =
1 + 7t4 + 11t6 + 85t8 + 217t10 + 782t12 + 1927t14 + 5127t16 + 11382t18 + 25351t20 + ...

### V9+V4

(1 + t + 2t2 + 2t3 + 4t4 + 7t5 + 16t6 + 35t7 + 79t8 + 167t9 + 330t10 + 620t11 + 1122t12 + 1949t13 + 3271t14 + 5326t15 + 8440t16 + 13035t17 + 19663t18 + 29005t19 + 41937t20 + 59465t21 + 82802t22 + 113312t23 + 152590t24 + 202274t25 + 264206t26 + 340189t27 + 432137t28 + 541721t29 + 670599t30 + 819961t31 + 990871t32 + 1183635t33 + 1398247t34 + 1633795t35 + 1888984t36 + 2161362t37 + 2448125t38 + 2745334t39 + 3048796t40 + 3353249t41 + 3653444t42 + 3943305t43 + 4217198t44 + 4468921t45 + 4693091t46 + 4884243t47 + 5038161t48 + 5150757t49 + 5219575t50 + 5242597t51 + 5219575t52 + 5150757t53 + 5038161t54 + 4884243t55 + 4693091t56 + 4468921t57 + 4217198t58 + 3943305t59 + 3653444t60 + 3353249t61 + 3048796t62 + 2745334t63 + 2448125t64 + 2161362t65 + 1888984t66 + 1633795t67 + 1398247t68 + 1183635t69 + 990871t70 + 819961t71 + 670599t72 + 541721t73 + 432137t74 + 340189t75 + 264206t76 + 202274t77 + 152590t78 + 113312t79 + 82802t80 + 59465t81 + 41937t82 + 29005t83 + 19663t84 + 13035t85 + 8440t86 + 5326t87 + 3271t88 + 1949t89 + 1122t90 + 620t91 + 330t92 + 167t93 + 79t94 + 35t95 + 16t96 + 7t97 + 4t98 + 2t99 + 2t100 + t101 + t102) / (1 + t)(1 + t2)(1 – t3)(1 – t5)(1 – t6)(1 – t7)(1 – t8)(1 – t9)(1 – t10)(1 – t11)(1 – t12)(1 – t13)(1 – t14)(1 – t16) =
1 + t2 + t3 + 3t4 + 5t5 + 12t6 + 24t7 + 54t8 + 108t9 + 206t10 + 382t11 + 697t12 + 1211t13 + 2060t14 + 3420t15 + 5558t16 + 8828t17 + 13780t18 + 21110t19 + 31873t20 + ...

### V9+V5

(1 + 6t4 + 22t6 + 160t8 + 575t10 + 2128t12 + 6124t14 + 16593t16 + 39298t18 + 87035t20 + 176195t22 + 336164t24 + 599145t26 + 1014210t28 + 1624585t30 + 2486843t32 + 3631423t34 + 5091264t36 + 6845690t38 + 8867492t40 + 11056036t42 + 13311360t44 + 15463205t46 + 17373740t48 + 18861406t50 + 19822754t52 + 20144015t54 + 19822754t56 + 18861406t58 + 17373740t60 + 15463205t62 + 13311360t64 + 11056036t66 + 8867492t68 + 6845690t70 + 5091264t72 + 3631423t74 + 2486843t76 + 1624585t78 + 1014210t80 + 599145t82 + 336164t84 + 176195t86 + 87035t88 + 39298t90 + 16593t92 + 6124t94 + 2128t96 + 575t98 + 160t100 + 22t102 + 6t104 + t108) / (1 – t4)2(1 – t6)2(1 – t8)2(1 – t10)2(1 – t12)2(1 – t14)2(1 – t16) =
1 + 8t4 + 24t6 + 177t8 + 637t10 + 2535t12 + 7756t14 + 23038t16 + 60057t18 + 149227t20 + ...

### V9+V6

(1 + t2 + 3t4 + 5t5 + 18t6 + 43t7 + 100t8 + 212t9 + 434t10 + 844t11 + 1576t12 + 2833t13 + 4924t14 + 8301t15 + 13584t16 + 21645t17 + 33648t18 + 51114t19 + 75969t20 + 110639t21 + 158079t22 + 221799t23 + 305914t24 + 415077t25 + 554532t26 + 729894t27 + 947158t28 + 1212367t29 + 1531644t30 + 1910571t31 + 2354290t32 + 2866796t33 + 3451058t34 + 4108135t35 + 4837506t36 + 5636151t37 + 6499086t38 + 7418448t39 + 8384261t40 + 9383736t41 + 10402355t42 + 11423127t43 + 12428066t44 + 13397698t45 + 14312638t46 + 15153120t47 + 15900815t48 + 16538336t49 + 17051031t50 + 17426322t51 + 17655350t52 + 17732211t53 + 17655350t54 + 17426322t55 + 17051031t56 + 16538336t57 + 15900815t58 + 15153120t59 + 14312638t60 + 13397698t61 + 12428066t62 + 11423127t63 + 10402355t64 + 9383736t65 + 8384261t66 + 7418448t67 + 6499086t68 + 5636151t69 + 4837506t70 + 4108135t71 + 3451058t72 + 2866796t73 + 2354290t74 + 1910571t75 + 1531644t76 + 1212367t77 + 947158t78 + 729894t79 + 554532t80 + 415077t81 + 305914t82 + 221799t83 + 158079t84 + 110639t85 + 75969t86 + 51114t87 + 33648t88 + 21645t89 + 13584t90 + 8301t91 + 4924t92 + 2833t93 + 1576t94 + 844t95 + 434t96 + 212t97 + 100t98 + 43t99 + 18t100 + 5t101 + 3t102 + t104 + t106) / (1 – t3)(1 – t4)(1 – t5)2(1 – t6)(1 – t7)(1 – t8)(1 – t9)(1 – t10)(1 – t11)(1 – t12)(1 – t13)(1 – t14)(1 – t16) =
1 + t2 + t3 + 4t4 + 8t5 + 21t6 + 50t7 + 114t8 + 248t9 + 522t10 + 1052t11 + 2050t12 + 3858t13 + 7059t14 + 12571t15 + 21834t16 + 37058t17 + 61615t18 + 100469t19 + 160914t20 + ...

### V9+V7

(1 + 9t4 + 34t6 + 316t8 + 1376t10 + 6145t12 + 21327t14 + 69016t16 + 195494t18 + 514758t20 + 1239720t22 + 2803268t24 + 5926539t26 + 11875242t28 + 22541559t30 + 40858085t32 + 70760999t34 + 117692307t36 + 188154477t38 + 290139577t40 + 431902205t42 + 622200146t44 + 868053204t46 + 1174979953t48 + 1543911973t50 + 1972062678t52 + 2449634251t54 + 2962218767t56 + 3488056675t58 + 4002604476t60 + 4476635223t62 + 4882745463t64 + 5193747679t66 + 5389906838t68 + 5456445653t70 + 5389906838t72 + 5193747679t74 + 4882745463t76 + 4476635223t78 + 4002604476t80 + 3488056675t82 + 2962218767t84 + 2449634251t86 + 1972062678t88 + 1543911973t90 + 1174979953t92 + 868053204t94 + 622200146t96 + 431902205t98 + 290139577t100 + 188154477t102 + 117692307t104 + 70760999t106 + 40858085t108 + 22541559t110 + 11875242t112 + 5926539t114 + 2803268t116 + 1239720t118 + 514758t120 + 195494t122 + 69016t124 + 21327t126 + 6145t128 + 1376t130 + 316t132 + 34t134 + 9t136 + t140) / (1 – t4)(1 – t6)2(1 – t8)2(1 – t10)3(1 – t12)3(1 – t14)2(1 – t16)2 =
1 + 10t4 + 36t6 + 328t8 + 1433t10 + 6565t12 + 23493t14 + 79132t16 + 235251t18 + 655183t20 + ...

### V9+V8

(1 – t + t2 – t3 + 3t4 + 5t5 + 18t6 + 44t7 + 107t8 + 243t9 + 516t10 + 1057t11 + 2077t12 + 3922t13 + 7159t14 + 12677t15 + 21787t16 + 36482t17 + 59595t18 + 95143t19 + 148699t20 + 227803t21 + 342491t22 + 505925t23 + 735012t24 + 1051072t25 + 1480738t26 + 2056566t27 + 2817780t28 + 3811012t29 + 5090779t30 + 6719732t31 + 8769051t32 + 11317917t33 + 14453216t34 + 18268551t35 + 22862859t36 + 28338357t37 + 34798649t38 + 42345444t39 + 51075560t40 + 61077255t41 + 72426474t42 + 85182277t43 + 99383558t44 + 115044175t45 + 132149850t46 + 150654704t47 + 170478823t48 + 191505914t49 + 213583526t50 + 236522177t51 + 260097506t52 + 284052550t53 + 308102006t54 + 331936416t55 + 355229483t56 + 377643843t57 + 398839592t58 + 418482173t59 + 436251120t60 + 451847548t61 + 465003684t62 + 475488659t63 + 483115967t64 + 487748305t65 + 489301963t66 + 487748305t67 + 483115967t68 + 475488659t69 + 465003684t70 + 451847548t71 + 436251120t72 + 418482173t73 + 398839592t74 + 377643843t75 + 355229483t76 + 331936416t77 + 308102006t78 + 284052550t79 + 260097506t80 + 236522177t81 + 213583526t82 + 191505914t83 + 170478823t84 + 150654704t85 + 132149850t86 + 115044175t87 + 99383558t88 + 85182277t89 + 72426474t90 + 61077255t91 + 51075560t92 + 42345444t93 + 34798649t94 + 28338357t95 + 22862859t96 + 18268551t97 + 14453216t98 + 11317917t99 + 8769051t100 + 6719732t101 + 5090779t102 + 3811012t103 + 2817780t104 + 2056566t105 + 1480738t106 + 1051072t107 + 735012t108 + 505925t109 + 342491t110 + 227803t111 + 148699t112 + 95143t113 + 59595t114 + 36482t115 + 21787t116 + 12677t117 + 7159t118 + 3922t119 + 2077t120 + 1057t121 + 516t122 + 243t123 + 107t124 + 44t125 + 18t126 + 5t127 + 3t128 – t129 + t130 – t131 + t132) / (1 – t)(1 – t3)(1 – t4)(1 – t5)(1 – t6)(1 – t7)(1 – t8)(1 – t9)(1 – t10)(1 – t11)(1 – t12)(1 – t13)(1 – t14)(1 – t15)(1 – t16)(1 – t17) =
1 + t2 + t3 + 4t4 + 10t5 + 29t6 + 76t7 + 193t8 + 463t9 + 1054t10 + 2298t11 + 4828t12 + 9769t13 + 19141t14 + 36420t15 + 67432t16 + 121786t17 + 214988t18 + 371546t19 + 629652t20 + ...

### V9+V9

(1 + t2 + 12t4 + 52t6 + 506t8 + 2686t10 + 13535t12 + 54909t14 + 201868t16 + 654947t18 + 1950166t20 + 5321784t22 + 13543268t24 + 32236286t26 + 72449952t28 + 154255943t30 + 312969394t32 + 606815392t34 + 1128798577t36 + 2019216018t38 + 3483144280t40 + 5804681043t42 + 9364866351t44 + 14647805695t46 + 22247078972t48 + 32847679743t50 + 47205725687t52 + 66091009102t54 + 90231865030t56 + 120216700984t58 + 156415867429t60 + 198865118538t62 + 247202018344t64 + 300577231985t66 + 357657523207t68 + 416613686709t70 + 475228980610t72 + 530984269068t74 + 581267659638t76 + 623526752304t78 + 655521907827t80 + 675470261849t82 + 682253718638t84 + 675470261849t86 + 655521907827t88 + 623526752304t90 + 581267659638t92 + 530984269068t94 + 475228980610t96 + 416613686709t98 + 357657523207t100 + 300577231985t102 + 247202018344t104 + 198865118538t106 + 156415867429t108 + 120216700984t110 + 90231865030t112 + 66091009102t114 + 47205725687t116 + 32847679743t118 + 22247078972t120 + 14647805695t122 + 9364866351t124 + 5804681043t126 + 3483144280t128 + 2019216018t130 + 1128798577t132 + 606815392t134 + 312969394t136 + 154255943t138 + 72449952t140 + 32236286t142 + 13543268t144 + 5321784t146 + 1950166t148 + 654947t150 + 201868t152 + 54909t154 + 13535t156 + 2686t158 + 506t160 + 52t162 + 12t164 + t166 + t168) / (1 – t4)(1 – t6)2(1 – t8)2(1 – t10)3(1 – t12)3(1 – t14)3(1 – t16)3 =
1 + t2 + 13t4 + 55t6 + 523t8 + 2770t10 + 14195t12 + 58844t14 + 222694t16 + 748175t18 + 2321390t20 + ...

### V10+V1

(1 + 3t + 4t2 + 3t3 + 2t4 + 2t5 + 5t6 + 12t7 + 27t8 + 59t9 + 116t10 + 201t11 + 332t12 + 521t13 + 772t14 + 1100t15 + 1525t16 + 2043t17 + 2670t18 + 3413t19 + 4257t20 + 5191t21 + 6211t22 + 7263t23 + 8312t24 + 9321t25 + 10234t26 + 10998t27 + 11589t28 + 11950t29 + 12069t30 + 11950t31 + 11589t32 + 10998t33 + 10234t34 + 9321t35 + 8312t36 + 7263t37 + 6211t38 + 5191t39 + 4257t40 + 3413t41 + 2670t42 + 2043t43 + 1525t44 + 1100t45 + 772t46 + 521t47 + 332t48 + 201t49 + 116t50 + 59t51 + 27t52 + 12t53 + 5t54 + 2t55 + 2t56 + 3t57 + 4t58 + 3t59 + t60) / (1 + t)3(1 – t3)(1 – t5)2(1 – t6)(1 – t7)2(1 – t8)(1 – t9)2(1 – t11) =
1 + t2 + 2t4 + t5 + 7t6 + 4t7 + 18t8 + 22t9 + 47t10 + 67t11 + 130t12 + 176t13 + 312t14 + 442t15 + 697t16 + 987t17 + 1498t18 + 2058t19 + 3005t20 + ...

### V10+V2

(1 + t + t2 + t4 – t5 + 11t6 + 16t7 + 44t8 + 70t9 + 131t10 + 178t11 + 294t12 + 385t13 + 549t14 + 693t15 + 919t16 + 1082t17 + 1348t18 + 1507t19 + 1730t20 + 1836t21 + 1979t22 + 1949t23 + 1979t24 + 1836t25 + 1730t26 + 1507t27 + 1348t28 + 1082t29 + 919t30 + 693t31 + 549t32 + 385t33 + 294t34 + 178t35 + 131t36 + 70t37 + 44t38 + 16t39 + 11t40 – t41 + t42 + t44 + t45 + t46) / (1 + t)(1 – t2)(1 – t3)(1 – t4)2(1 – t5)2(1 – t6)2(1 – t7)(1 – t8)(1 – t9) =
1 + 2t2 + 6t4 + 24t6 + 7t7 + 74t8 + 55t9 + 220t10 + 215t11 + 621t12 + 694t13 + 1589t14 + 1947t15 + 3810t16 + 4852t17 + 8606t18 + 11125t19 + 18308t20 + ...

### V10+V3

(1 + 2t + 2t2 + t3 + t4 + 3t5 + 11t6 + 29t7 + 68t8 + 144t9 + 281t10 + 512t11 + 895t12 + 1494t13 + 2402t14 + 3739t15 + 5652t16 + 8297t17 + 11873t18 + 16568t19 + 22578t20 + 30078t21 + 39218t22 + 50068t23 + 62667t24 + 76933t25 + 92709t26 + 109721t27 + 127613t28 + 145894t29 + 164055t30 + 181484t31 + 197576t32 + 211723t33 + 223391t34 + 232074t35 + 237448t36 + 239262t37 + 237448t38 + 232074t39 + 223391t40 + 211723t41 + 197576t42 + 181484t43 + 164055t44 + 145894t45 + 127613t46 + 109721t47 + 92709t48 + 76933t49 + 62667t50 + 50068t51 + 39218t52 + 30078t53 + 22578t54 + 16568t55 + 11873t56 + 8297t57 + 5652t58 + 3739t59 + 2402t60 + 1494t61 + 895t62 + 512t63 + 281t64 + 144t65 + 68t66 + 29t67 + 11t68 + 3t69 + t70 + t71 + 2t72 + 2t73 + t74) / (1 + t)2(1 – t3)(1 – t4)(1 – t5)2(1 – t6)(1 – t7)2(1 – t8)(1 – t9)2(1 – t11)(1 – t13) =
1 + t2 + 3t4 + 3t5 + 11t6 + 17t7 + 43t8 + 79t9 + 158t10 + 284t11 + 525t12 + 897t13 + 1544t14 + 2540t15 + 4141t16 + 6549t17 + 10253t18 + 15662t19 + 23673t20 + ...

### V10+V4

(1 + t + t2 + t3 + 2t4 + 7t5 + 27t6 + 54t7 + 118t8 + 225t9 + 409t10 + 695t11 + 1170t12 + 1822t13 + 2789t14 + 4072t15 + 5755t16 + 7806t17 + 10320t18 + 13090t19 + 16200t20 + 19387t21 + 22573t22 + 25474t23 + 28073t24 + 29958t25 + 31236t26 + 31631t27 + 31236t28 + 29958t29 + 28073t30 + 25474t31 + 22573t32 + 19387t33 + 16200t34 + 13090t35 + 10320t36 + 7806t37 + 5755t38 + 4072t39 + 2789t40 + 1822t41 + 1170t42 + 695t43 + 409t44 + 225t45 + 118t46 + 54t47 + 27t48 + 7t49 + 2t50 + t51 + t52 + t53 + t54) / (1 + t)(1 – t2)(1 – t3)2(1 – t4)2(1 – t5)2(1 – t6)2(1 – t7)2(1 – t8)(1 – t9) =
1 + 2t2 + 2t3 + 6t4 + 11t5 + 35t6 + 55t7 + 148t8 + 271t9 + 571t10 + 1048t11 + 2037t12 + 3537t13 + 6403t14 + 10778t15 + 18285t16 + 29715t17 + 48233t18 + 75624t19 + 118229t20 + ...

### V10+V5

(1 + t – t3 + 4t5 + 17t6 + 42t7 + 94t8 + 196t9 + 384t10 + 713t11 + 1279t12 + 2191t13 + 3625t14 + 5805t15 + 9004t16 + 13544t17 + 19834t18 + 28267t19 + 39297t20 + 53362t21 + 70855t22 + 92067t23 + 117236t24 + 146347t25 + 179265t26 + 215607t27 + 254759t28 + 295837t29 + 337850t30 + 379501t31 + 419478t32 + 456375t33 + 488825t34 + 515521t35 + 535466t36 + 547754t37 + 551912t38 + 547754t39 + 535466t40 + 515521t41 + 488825t42 + 456375t43 + 419478t44 + 379501t45 + 337850t46 + 295837t47 + 254759t48 + 215607t49 + 179265t50 + 146347t51 + 117236t52 + 92067t53 + 70855t54 + 53362t55 + 39297t56 + 28267t57 + 19834t58 + 13544t59 + 9004t60 + 5805t61 + 3625t62 + 2191t63 + 1279t64 + 713t65 + 384t66 + 196t67 + 94t68 + 42t69 + 17t70 + 4t71 – t73 + t75 + t76) / (1 + t)(1 – t2)(1 – t3)2(1 – t4)(1 – t5)2(1 – t6)(1 – t7)2(1 – t8)(1 – t9)2(1 – t11)(1 – t13) =
1 + t2 + t3 + 3t4 + 6t5 + 19t6 + 39t7 + 96t8 + 208t9 + 437t10 + 880t11 + 1726t12 + 3233t13 + 5934t14 + 10557t15 + 18334t16 + 31114t17 + 51753t18 + 84324t19 + 135081t20 + ...

### V10+V6

(1 + 2t + 2t2 + 4t4 + 14t5 + 60t6 + 155t7 + 388t8 + 832t9 + 1755t10 + 3393t11 + 6400t12 + 11370t13 + 19567t14 + 32128t15 + 51181t16 + 78317t17 + 116364t18 + 167016t19 + 233164t20 + 315582t21 + 416199t22 + 533527t23 + 667304t24 + 812910t25 + 967211t26 + 1122325t27 + 1273057t28 + 1409574t29 + 1526396t30 + 1614478t31 + 1670616t32 + 1688923t33 + 1670616t34 + 1614478t35 + 1526396t36 + 1409574t37 + 1273057t38 + 1122325t39 + 967211t40 + 812910t41 + 667304t42 + 533527t43 + 416199t44 + 315582t45 + 233164t46 + 167016t47 + 116364t48 + 78317t49 + 51181t50 + 32128t51 + 19567t52 + 11370t53 + 6400t54 + 3393t55 + 1755t56 + 832t57 + 388t58 + 155t59 + 60t60 + 14t61 + 4t62 + 2t64 + 2t65 + t66) / (1 + t)2(1 – t2)(1 – t3)2(1 – t4)2(1 – t5)3(1 – t6)2(1 – t7)2(1 – t8)2(1 – t9) =
1 + 2t2 + 11t4 + 7t5 + 63t6 + 81t7 + 320t8 + 534t9 + 1476t10 + 2621t11 + 6019t12 + 10702t13 + 21762t14 + 37966t15 + 71017t16 + 120355t17 + 211705t18 + 348131t19 + 583391t20 + ...

### V10+V7

(1 – t + t2 – t3 + 2t4 + 5t5 + 18t6 + 37t7 + 105t8 + 221t9 + 480t10 + 983t11 + 1928t12 + 3609t13 + 6622t14 + 11637t15 + 19952t16 + 33279t17 + 54091t18 + 85817t19 + 133374t20 + 202713t21 + 302344t22 + 442612t23 + 636604t24 + 900389t25 + 1253743t26 + 1718916t27 + 2323144t28 + 3096261t29 + 4071815t30 + 5286224t31 + 6778906t32 + 8588855t33 + 10757651t34 + 13323804t35 + 16323734t36 + 19788972t37 + 23745618t38 + 28208564t39 + 33185927t40 + 38670767t41 + 44643866t42 + 51070632t43 + 57902631t44 + 65072032t45 + 72500566t46 + 80091705t47 + 87738207t48 + 95321458t49 + 102717391t50 + 109792927t51 + 116420161t52 + 122469447t53 + 127820577t54 + 132362959t55 + 136002973t56 + 138659457t57 + 140278038t58 + 140821318t59 + 140278038t60 + 138659457t61 + 136002973t62 + 132362959t63 + 127820577t64 + 122469447t65 + 116420161t66 + 109792927t67 + 102717391t68 + 95321458t69 + 87738207t70 + 80091705t71 + 72500566t72 + 65072032t73 + 57902631t74 + 51070632t75 + 44643866t76 + 38670767t77 + 33185927t78 + 28208564t79 + 23745618t80 + 19788972t81 + 16323734t82 + 13323804t83 + 10757651t84 + 8588855t85 + 6778906t86 + 5286224t87 + 4071815t88 + 3096261t89 + 2323144t90 + 1718916t91 + 1253743t92 + 900389t93 + 636604t94 + 442612t95 + 302344t96 + 202713t97 + 133374t98 + 85817t99 + 54091t100 + 33279t101 + 19952t102 + 11637t103 + 6622t104 + 3609t105 + 1928t106 + 983t107 + 480t108 + 221t109 + 105t110 + 37t111 + 18t112 + 5t113 + 2t114 – t115 + t116 – t117 + t118) / (1 – t)(1 – t3)(1 – t4)(1 – t5)(1 – t6)(1 – t7)2(1 – t8)(1 – t9)2(1 – t10)(1 – t11)(1 – t12)(1 – t13)(1 – t15)(1 – t17) =
1 + t2 + t3 + 3t4 + 9t5 + 28t6 + 68t7 + 181t8 + 430t9 + 975t10 + 2137t11 + 4490t12 + 9061t13 + 17786t14 + 33824t15 + 62592t16 + 113065t17 + 199573t18 + 344793t19 + 584342t20 + ...

### V10+V8

(1 + 2t + 2t2 + 2t3 + 8t4 + 28t5 + 97t6 + 267t7 + 689t8 + 1663t9 + 3806t10 + 8195t11 + 16895t12 + 33202t13 + 62611t14 + 113476t15 + 198368t16 + 334793t17 + 547296t18 + 867446t19 + 1335661t20 + 2000308t21 + 2917945t22 + 4149556t23 + 5759695t24 + 7808673t25 + 10349346t26 + 13417558t27 + 17027798t28 + 21162440t29 + 25771512t30 + 30763670t31 + 36011454t32 + 41349765t33 + 46588015t34 + 51514158t35 + 55916021t36 + 59587657t37 + 62352174t38 + 64069124t39 + 64652347t40 + 64069124t41 + 62352174t42 + 59587657t43 + 55916021t44 + 51514158t45 + 46588015t46 + 41349765t47 + 36011454t48 + 30763670t49 + 25771512t50 + 21162440t51 + 17027798t52 + 13417558t53 + 10349346t54 + 7808673t55 + 5759695t56 + 4149556t57 + 2917945t58 + 2000308t59 + 1335661t60 + 867446t61 + 547296t62 + 334793t63 + 198368t64 + 113476t65 + 62611t66 + 33202t67 + 16895t68 + 8195t69 + 3806t70 + 1663t71 + 689t72 + 267t73 + 97t74 + 28t75 + 8t76 + 2t77 + 2t78 + 2t79 + t80) / (1 + t)2(1 – t2)(1 – t3)2(1 – t4)2(1 – t5)3(1 – t6)2(1 – t7)3(1 – t8)2(1 – t9)2 =
1 + 2t2 + 2t3 + 11t4 + 21t5 + 80t6 + 174t7 + 507t8 + 1166t9 + 2844t10 + 6223t11 + 13747t12 + 28279t13 + 57704t14 + 112381t15 + 215280t16 + 398794t17 + 725942t18 + 1286867t19 + 2243282t20 + ...

### V10+V9

(1 – t + t2 – t3 + 3t4 + 6t5 + 26t6 + 65t7 + 177t8 + 429t9 + 1001t10 + 2214t11 + 4704t12 + 9570t13 + 18834t14 + 35828t15 + 66126t16 + 118692t17 + 207617t18 + 354450t19 + 591773t20 + 967357t21 + 1550407t22 + 2439133t23 + 3770633t24 + 5732998t25 + 8580890t26 + 12653081t27 + 18394627t28 + 26381781t29 + 37350789t30 + 52229781t31 + 72174974t32 + 98607400t33 + 133253597t34 + 178185530t35 + 235860100t36 + 309155912t37 + 401406870t38 + 516426143t39 + 658524094t40 + 832512406t41 + 1043694118t42 + 1297836635t43 + 1601127192t44 + 1960103373t45 + 2381565155t46 + 2872461159t47 + 3439751272t48 + 4090247359t49 + 4830434470t50 + 5666269697t51 + 6602972095t52 + 7644800626t53 + 8794827963t54 + 10054719524t55 + 11424523619t56 + 12902475779t57 + 14484834598t58 + 16165749107t59 + 17937165599t60 + 19788785784t61 + 21708077649t62 + 23680339805t63 + 25688831269t64 + 27714959742t65 + 29738525788t66 + 31738028353t67 + 33691018369t68 + 35574492374t69 + 37365325147t70 + 39040723657t71 + 40578688259t72 + 41958479096t73 + 43161064247t74 + 44169537975t75 + 44969503599t76 + 45549403285t77 + 45900781124t78 + 46018483899t79 + 45900781124t80 + 45549403285t81 + 44969503599t82 + 44169537975t83 + 43161064247t84 + 41958479096t85 + 40578688259t86 + 39040723657t87 + 37365325147t88 + 35574492374t89 + 33691018369t90 + 31738028353t91 + 29738525788t92 + 27714959742t93 + 25688831269t94 + 23680339805t95 + 21708077649t96 + 19788785784t97 + 17937165599t98 + 16165749107t99 + 14484834598t100 + 12902475779t101 + 11424523619t102 + 10054719524t103 + 8794827963t104 + 7644800626t105 + 6602972095t106 + 5666269697t107 + 4830434470t108 + 4090247359t109 + 3439751272t110 + 2872461159t111 + 2381565155t112 + 1960103373t113 + 1601127192t114 + 1297836635t115 + 1043694118t116 + 832512406t117 + 658524094t118 + 516426143t119 + 401406870t120 + 309155912t121 + 235860100t122 + 178185530t123 + 133253597t124 + 98607400t125 + 72174974t126 + 52229781t127 + 37350789t128 + 26381781t129 + 18394627t130 + 12653081t131 + 8580890t132 + 5732998t133 + 3770633t134 + 2439133t135 + 1550407t136 + 967357t137 + 591773t138 + 354450t139 + 207617t140 + 118692t141 + 66126t142 + 35828t143 + 18834t144 + 9570t145 + 4704t146 + 2214t147 + 1001t148 + 429t149 + 177t150 + 65t151 + 26t152 + 6t153 + 3t154 – t155 + t156 – t157 + t158) / (1 – t)(1 – t3)(1 – t4)(1 – t5)(1 – t6)(1 – t7)(1 – t8)(1 – t9)2(1 – t10)(1 – t11)(1 – t12)(1 – t13)(1 – t14)(1 – t15)(1 – t16)(1 – t17)(1 – t19) =
1 + t2 + t3 + 4t4 + 11t5 + 38t6 + 106t7 + 294t8 + 760t9 + 1865t10 + 4368t11 + 9820t12 + 21214t13 + 44306t14 + 89644t15 + 176177t16 + 337163t17 + 629652t18 + 1149381t19 + 2054420t20 + ...

### V10+V10

(1 + 2t + 3t2 + 3t3 + 9t4 + 28t5 + 124t6 + 371t7 + 1099t8 + 2933t9 + 7437t10 + 17556t11 + 39866t12 + 85888t13 + 177959t14 + 354088t15 + 680345t16 + 1262224t17 + 2270582t18 + 3961664t19 + 6721578t20 + 11099600t21 + 17870427t22 + 28073601t23 + 43092034t24 + 64674562t25 + 95006558t26 + 136690962t27 + 192770576t28 + 266618338t29 + 361889200t30 + 482274948t31 + 631359401t32 + 812253396t33 + 1027368940t34 + 1277968789t35 + 1563969796t36 + 1883498234t37 + 2232827783t38 + 2606108309t39 + 2995553498t40 + 3391420781t41 + 3782560838t42 + 4156660715t43 + 4501080490t44 + 4803314850t45 + 5051952624t46 + 5237122666t47 + 5351399696t48 + 5390006050t49 + 5351399696t50 + 5237122666t51 + 5051952624t52 + 4803314850t53 + 4501080490t54 + 4156660715t55 + 3782560838t56 + 3391420781t57 + 2995553498t58 + 2606108309t59 + 2232827783t60 + 1883498234t61 + 1563969796t62 + 1277968789t63 + 1027368940t64 + 812253396t65 + 631359401t66 + 482274948t67 + 361889200t68 + 266618338t69 + 192770576t70 + 136690962t71 + 95006558t72 + 64674562t73 + 43092034t74 + 28073601t75 + 17870427t76 + 11099600t77 + 6721578t78 + 3961664t79 + 2270582t80 + 1262224t81 + 680345t82 + 354088t83 + 177959t84 + 85888t85 + 39866t86 + 17556t87 + 7437t88 + 2933t89 + 1099t90 + 371t91 + 124t92 + 28t93 + 9t94 + 3t95 + 3t96 + 2t97 + t98) / (1 + t)2(1 – t2)(1 – t3)(1 – t4)2(1 – t5)3(1 – t6)3(1 – t7)3(1 – t8)3(1 – t9)3 =
1 + 3t2 + 14t4 + 16t5 + 114t6 + 208t7 + 815t8 + 1812t9 + 5167t10 + 11554t11 + 28334t12 + 60688t13 + 134326t14 + 274312t15 + 562638t16 + 1095422t17 + 2118218t18 + 3945746t19 + 7268700t20 + ...