Invariants for one form, for several forms.

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

# The Poincaré series for multiple 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 several (more than 2) 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.

## Three forms

### V1+V1+V1

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

Multivariate: 1 / (1 – tu)(1 – tv)(1 – uv)

### V2+V1+V1

(1 + t3) / (1 – t2)2(1 – t3)2 =
1 + 2t2 + 3t3 + 3t4 + 6t5 + 9t6 + 9t7 + 15t8 + 19t9 + 21t10 + 29t11 + 36t12 + 39t13 + 51t14 + 60t15 + 66t16 + 81t17 + 94t18 + 102t19 + 122t20 + ...

Multivariate: (1 + tuv) / (1 – tu)(1 – v2)(1 – t2v)(1 – u2v)

### V2+V2+V1

(1 + t4) / (1 – t2)3(1 – t3)2 =
1 + 3t2 + 2t3 + 7t4 + 6t5 + 16t6 + 14t7 + 30t8 + 30t9 + 52t10 + 54t11 + 87t12 + 90t13 + 135t14 + 144t15 + 201t16 + 216t17 + 292t18 + 312t19 + 408t20 + ...

Multivariate: (1 + t2uv) / (1 – u2)(1 – uv)(1 – v2)(1 – t2u)(1 – t2v)

### V2+V2+V2

(1 + t3) / (1 – t2)6 =
1 + 6t2 + t3 + 21t4 + 6t5 + 56t6 + 21t7 + 126t8 + 56t9 + 252t10 + 126t11 + 462t12 + 252t13 + 792t14 + 462t15 + 1287t16 + 792t17 + 2002t18 + 1287t19 + 3003t20 + ...

Multivariate: (1 + tuv) / (1 – t2)(1 – tu)(1 – u2)(1 – tv)(1 – uv)(1 – v2)

### V3+V1+V1

(1 + 4t4 + 4t6 + t10) / (1 – t2)(1 – t4)4 =
(1 – t2 + 5t4 – t6 + t8) / (1 – t2)2(1 – t4)3 =
1 + t2 + 9t4 + 13t6 + 39t8 + 56t10 + 116t12 + 160t14 + 275t16 + 365t18 + 561t20 + ...

Multivariate: (1 – tv – uv + t2uv + tu2v + t2v2 + 2tuv2 + u2v2 – t3uv2 – 2t2u2v2 – tu3v2 – t2uv3 – tu2v3 + t3u2v3 + t2u3v3 – t3u3v4) / (1 – tu)(1 – tv)(1 – uv)(1 – t3v)(1 – u3v)(1 – v4)

### V3+V2+V1

(1 + t3 + 2t4 + 3t5 + 3t6 + 3t7 + 2t8 + t9 + t12) / (1 – t2)(1 – t3)2(1 – t4)2(1 – t5) =
1 + t2 + 3t3 + 5t4 + 7t5 + 13t6 + 20t7 + 31t8 + 44t9 + 63t10 + 88t11 + 123t12 + 160t13 + 213t14 + 279t15 + 358t16 + 450t17 + 569t18 + 705t19 + 871t20 + ...

Multivariate: (1 – tv + tuv + tu2v + t2v2 + t3uv + tuv3 + tu2v3 – t4uv2 + t2u3v2 – t2uv4 + u3v4 – t3u2v3 – t3u3v3 – tu3v5 – t2u4v4 – t3u2v5 – t3u3v5 + t3u4v5 – t4u4v6) / (1 – u2)(1 – tv)(1 – t2u)(1 – uv2)(1 – t3v)(1 – v4)(1 – u3v2)

### V3+V2+V2

(1 + t2 + 2t4 + 2t5 + 5t6 + 6t7 + 5t8 + 2t9 + 2t10 + t12 + t14) / (1 – t2)2(1 – t3)2(1 – t4)(1 – t5)2 =
1 + 3t2 + 2t3 + 8t4 + 10t5 + 22t6 + 32t7 + 55t8 + 80t9 + 128t10 + 178t11 + 268t12 + 362t13 + 515t14 + 686t15 + 933t16 + 1218t17 + 1605t18 + 2054t19 + 2644t20 + ...

Multivariate: (1 + tuv2 + t2uv2 + tu2v2 + t3uv2 + t2u2v2 + tu3v2 + t3v4 + t2uv4 + tu2v4 + u3v4 – t4uv4 – t3u2v4 – t2u3v4 – tu4v4 – t3uv6 – t2u2v6 – tu3v6 – t3u2v6 – t2u3v6 – t3u3v6 – t4u4v8) / (1 – t2)(1 – tu)(1 – u2)(1 – tv2)(1 – uv2)(1 – v4)(1 – t3v2)(1 – u3v2)

### V3+V3+V1

(1 + 6t4 + 13t6 + 12t8 + 13t10 + 6t12 + t16) / (1 – t2)(1 – t4)6 =
(1 – 2t2 + 9t4 – 3t6 + 9t8 – 2t10 + t12) / (1 – t2)3(1 – t4)4 =
1 + t2 + 13t4 + 26t6 + 95t8 + 186t10 + 446t12 + 797t14 + 1548t16 + 2549t18 + 4361t20 + ...

Multivariate: (1 – tu – tv + t2u2 + 2t2uv + tu2v + t2v2 + tuv2 + u2v2 + t4uv – t3u2v + tu4v – t3uv2 – t2u2v2 + u3v3 + tuv4 – t5u2v – t3u4v – t2u5v – t5uv2 – t2u4v2 – t3uv4 – t2u2v4 – t2uv5 + t6u2v2 + t3u5v2 – t3u4v3 – t3u3v4 + t3u2v5 + u5v5 – t4u6v2 – t4u5v3 – t3u6v3 – t4u3v5 – tu6v5 – t4u2v6 – t3u3v6 – tu5v6 + t5u6v3 + t6u4v4 – t4u5v5 – t3u6v5 + t5u3v6 – t3u5v6 + t2u6v6 + t6u5v5 + t5u6v5 + t4u7v5 + t5u5v6 + 2t4u6v6 + t4u5v7 – t5u7v6 – t5u6v7 + t6u7v7) / (1 – tu)(1 – tv)(1 – uv)(1 – t3u)(1 – u4)(1 – t3v)(1 – u3v)(1 – uv3)(1 – v4)

### V3+V3+V2

(1 + t + t2 + 2t3 + 5t4 + 9t5 + 14t6 + 22t7 + 29t8 + 32t9 + 34t10 + 32t11 + 29t12 + 22t13 + 14t14 + 9t15 + 5t16 + 2t17 + t18 + t19 + t20) / (1 + t)(1 – t2)(1 – t3)2(1 – t4)3(1 – t5)2 =
1 + 2t2 + 3t3 + 9t4 + 12t5 + 26t6 + 44t7 + 79t8 + 120t9 + 200t10 + 300t11 + 467t12 + 666t13 + 982t14 + 1377t15 + 1946t16 + 2639t17 + 3613t18 + 4806t19 + 6406t20 + ...

Multivariate: (1 + tuv + t2uv + u2v2 + t3uv + tu3v + tu2v2 + tuv3 + t2u3v + 2t2u2v2 + t2uv3 + u3v3 + t3u4 + t3u3v + 2t3u2v2 + t3uv3 + tu3v3 + t3v4 + t4u2v2 + t2u4v2 + t2u3v3 + t2u2v4 – t3u5v + t5u2v2 – t3u4v2 + t3u3v3 – t3u2v4 + tu4v4 – t3uv5 – t2u6v2 – t2u5v3 – t2u3v5 + u5v5 – t2u2v6 – t3u7v – t3u6v2 – 2t3u5v3 – 2t3u4v4 – 2t3u3v5 – t3u2v6 – t3uv7 – t6u4v2 – t4u6v2 – t4u5v3 – t2u7v3 – t6u2v4 + t4u4v4 – t2u6v4 – t4u3v5 – t2u5v5 – t4u2v6 – t2u4v6 – t2u3v7 – t5u6v2 – t5u5v3 – t3u7v3 – t5u4v4 – t3u6v4 – t5u3v5 + t3u5v5 – tu7v5 – t5u2v6 – t3u4v6 – t3u3v7 – tu5v7 – t4u8v2 – t4u7v3 – 2t4u6v4 – 2t4u5v5 – 2t4u4v6 – t4u3v7 – t4u2v8 – t5u7v3 + t7u4v4 – t5u6v4 – t5u4v6 – t5u3v7 – t4u8v4 + t6u5v5 – t4u7v5 + t4u6v6 – t4u5v7 + t2u7v7 – t4u4v8 + t5u7v5 + t5u6v6 + t5u5v7 + t3u7v7 + t4u9v5 + t6u6v6 + t4u8v6 + 2t4u7v7 + t4u6v8 + t4u5v9 + t7u6v6 + t5u8v6 + 2t5u7v7 + t5u6v8 + t6u8v6 + t6u7v7 + t6u6v8 + t4u8v8 + t7u7v7 + t5u8v8 + t6u8v8 + t7u9v9) / (1 – t2)(1 – uv)(1 – tu2)(1 – tv2)(1 – u4)(1 – u3v)(1 – uv3)(1 – v4)(1 – t3u2)(1 – t3v2)

### V3+V3+V3

(1 – t2 + 10t4 + 30t8 + 10t12 – t14 + t16) / (1 – t2)4(1 – t4)5 =
1 + 3t2 + 21t4 + 65t6 + 240t8 + 636t10 + 1688t12 + 3795t14 + 8277t16 + 16379t18 + 31377t20 + ...

Multivariate: (1 + t2u2 + t2uv + tu2v + t2v2 + tuv2 + u2v2 + t3u3 + t3u2v + t2u3v + t3uv2 + t2u2v2 + tu3v2 + t3v3 + t2uv3 + tu2v3 + u3v3 + t4u2v2 + t3u3v2 + t2u4v2 + t3u2v3 + t2u3v3 + t2u2v4 + t5u5 – t6u3v + t5u4v + t4u5v – t3u6v – t6u2v2 + t4u4v2 – t2u6v2 – t6uv3 – t4u3v3 – t3u4v3 – tu6v3 + t5uv4 + t4u2v4 – t3u3v4 + t2u4v4 + tu5v4 + t5v5 + t4uv5 + tu4v5 + u5v5 – t3uv6 – t2u2v6 – tu3v6 – t6u5v – t5u6v – t8u2v2 – t7u3v2 – 2t6u4v2 – 3t5u5v2 – 2t4u6v2 – t3u7v2 – t2u8v2 – t7u2v3 – 2t6u3v3 – 2t5u4v3 – 2t4u5v3 – 2t3u6v3 – t2u7v3 – 2t6u2v4 – 2t5u3v4 – 2t3u5v4 – 2t2u6v4 – t6uv5 – 3t5u2v5 – 2t4u3v5 – 2t3u4v5 – 3t2u5v5 – tu6v5 – t5uv6 – 2t4u2v6 – 2t3u3v6 – 2t2u4v6 – tu5v6 – t3u2v7 – t2u3v7 – t2u2v8 – t8u5v – t5u8v – t8u4v2 – t7u5v2 + t6u6v2 – t5u7v2 – t4u8v2 – t8u3v3 – t7u4v3 – t6u5v3 – t5u6v3 – t4u7v3 – t3u8v3 – t8u2v4 – t7u3v4 – 2t6u4v4 – 5t5u5v4 – 2t4u6v4 – t3u7v4 – t2u8v4 – t8uv5 – t7u2v5 – t6u3v5 – 5t5u4v5 – 5t4u5v5 – t3u6v5 – t2u7v5 – tu8v5 + t6u2v6 – t5u3v6 – 2t4u4v6 – t3u5v6 + t2u6v6 – t5u2v7 – t4u3v7 – t3u4v7 – t2u5v7 – t5uv8 – t4u2v8 – t3u3v8 – t2u4v8 – tu5v8 + t9u5v2 + t8u6v2 + t7u7v2 + t6u8v2 + t5u9v2 + t8u5v3 + t7u6v3 + t6u7v3 + t5u8v3 – t8u4v4 + t7u5v4 + 2t6u6v4 + t5u7v4 – t4u8v4 + t9u2v5 + t8u3v5 + t7u4v5 + 5t6u5v5 + 5t5u6v5 + t4u7v5 + t3u8v5 + t2u9v5 + t8u2v6 + t7u3v6 + 2t6u4v6 + 5t5u5v6 + 2t4u6v6 + t3u7v6 + t2u8v6 + t7u2v7 + t6u3v7 + t5u4v7 + t4u5v7 + t3u6v7 + t2u7v7 + t6u2v8 + t5u3v8 – t4u4v8 + t3u5v8 + t2u6v8 + t5u2v9 + t2u5v9 + t8u8v2 + t8u7v3 + t7u8v3 + t9u5v4 + 2t8u6v4 + 2t7u7v4 + 2t6u8v4 + t5u9v4 + t9u4v5 + 3t8u5v5 + 2t7u6v5 + 2t6u7v5 + 3t5u8v5 + t4u9v5 + 2t8u4v6 + 2t7u5v6 + 2t5u7v6 + 2t4u8v6 + t8u3v7 + 2t7u4v7 + 2t6u5v7 + 2t5u6v7 + 2t4u7v7 + t3u8v7 + t8u2v8 + t7u3v8 + 2t6u4v8 + 3t5u5v8 + 2t4u6v8 + t3u7v8 + t2u8v8 + t5u4v9 + t4u5v9 + t9u7v4 + t8u8v4 + t7u9v4 – t10u5v5 – t9u6v5 – t6u9v5 – t5u10v5 – t9u5v6 – t8u6v6 + t7u7v6 – t6u8v6 – t5u9v6 + t9u4v7 + t7u6v7 + t6u7v7 + t4u9v7 + t8u4v8 – t6u6v8 + t4u8v8 + t7u4v9 – t6u5v9 – t5u6v9 + t4u7v9 – t5u5v10 – t8u8v6 – t8u7v7 – t7u8v7 – t8u6v8 – t7u7v8 – t6u8v8 – t10u7v7 – t9u8v7 – t8u9v7 – t7u10v7 – t9u7v8 – t8u8v8 – t7u9v8 – t8u7v9 – t7u8v9 – t7u7v10 – t10u8v8 – t9u9v8 – t8u10v8 – t9u8v9 – t8u9v9 – t8u8v10 – t10u10v10) / (1 – tu)(1 – tv)(1 – uv)(1 – t4)(1 – t3u)(1 – tu3)(1 – u4)(1 – t3v)(1 – u3v)(1 – tv3)(1 – uv3)(1 – v4)

### V4+V1+V1

(1 + t2 + t4 + 3t5 + 4t6 + 3t7 + 4t8 + 10t9 + 4t10 + 3t11 + 4t12 + 3t13 + t14 + t16 + t18) / (1 – t2)(1 – t3)(1 – t5)2(1 – t6)2 =
(1 + t2 – 2t3 + t4 + t5 + 7t6 + t7 + t8 – 2t9 + t10 + t12) / (1 – t2)(1 – t3)3(1 – t5)2 =
1 + 2t2 + t3 + 3t4 + 7t5 + 10t6 + 13t7 + 22t8 + 32t9 + 43t10 + 65t11 + 86t12 + 107t13 + 154t14 + 195t15 + 244t16 + 321t17 + 393t18 + 484t19 + 612t20 + ...

### V4+V2+V1

(1 + t4 + 2t5 + 4t6 + 2t7 + t8 + t12) / (1 – t2)2(1 – t3)3(1 – t4)(1 – t5) =
(1 – t2 + 2t4 + 2t5 + 2t6 – t8 + t10) / (1 – t2)3(1 – t3)3(1 – t5) =
1 + 2t2 + 3t3 + 5t4 + 9t5 + 18t6 + 23t7 + 43t8 + 63t9 + 93t10 + 136t11 + 200t12 + 267t13 + 381t14 + 510t15 + 679t16 + 896t17 + 1177t18 + 1494t19 + 1928t20 + ...

### V4+V2+V2

(1 + t3 + 4t4 + 2t5 + 4t6 + t7 + t10) / (1 – t2)4(1 – t3)3(1 – t4) =
(1 – t2 + t3 + 5t4 + t5 – t6 + t8) / (1 – t2)5(1 – t3)3 =
1 + 4t2 + 4t3 + 15t4 + 18t5 + 53t6 + 65t7 + 148t8 + 198t9 + 371t10 + 499t11 + 853t12 + 1129t13 + 1786t14 + 2354t15 + 3498t16 + 4549t17 + 6487t18 + 8303t19 + 11429t20 + ...

### V4+V3+V1

(1 + t + t2 + 2t4 + 8t5 + 18t6 + 28t7 + 36t8 + 43t9 + 47t10 + 50t11 + 51t12 + 50t13 + 47t14 + 43t15 + 36t16 + 28t17 + 18t18 + 8t19 + 2t20 + t22 + t23 + t24) / (1 + t)(1 – t3)3(1 – t4)2(1 – t5)2(1 – t7) =
1 + t2 + 2t3 + 5t4 + 10t5 + 18t6 + 31t7 + 55t8 + 92t9 + 144t10 + 223t11 + 341t12 + 499t13 + 725t14 + 1031t15 + 1436t16 + 1978t17 + 2685t18 + 3592t19 + 4761t20 + ...

### V4+V3+V2

(1 + 3t4 + 6t5 + 11t6 + 14t7 + 17t8 + 19t9 + 21t10 + 20t11 + 21t12 + 19t13 + 17t14 + 14t15 + 11t16 + 6t17 + 3t18 + t22) / (1 – t2)2(1 – t3)3(1 – t4)(1 – t5)2(1 – t7) =
1 + 2t2 + 3t3 + 7t4 + 14t5 + 29t6 + 52t7 + 96t8 + 166t9 + 279t10 + 451t11 + 720t12 + 1104t13 + 1673t14 + 2475t15 + 3596t16 + 5137t17 + 7238t18 + 10036t19 + 13759t20 + ...

### V4+V3+V3

(1 + 2t + 3t2 + 3t3 + 6t4 + 19t5 + 50t6 + 105t7 + 188t8 + 300t9 + 439t10 + 598t11 + 774t12 + 966t13 + 1164t14 + 1347t15 + 1479t16 + 1526t17 + 1479t18 + 1347t19 + 1164t20 + 966t21 + 774t22 + 598t23 + 439t24 + 300t25 + 188t26 + 105t27 + 50t28 + 19t29 + 6t30 + 3t31 + 3t32 + 2t33 + t34) / (1 + t)2(1 – t3)3(1 – t4)3(1 – t5)2(1 – t7)2 =
1 + 2t2 + 2t3 + 9t4 + 16t5 + 37t6 + 71t7 + 142t8 + 255t9 + 454t10 + 768t11 + 1287t12 + 2067t13 + 3264t14 + 5020t15 + 7600t16 + 11258t17 + 16443t18 + 23606t19 + 33474t20 + ...

### V4+V4+V1

(1 + 2t4 + 2t5 + 6t6 + 4t7 + 4t8 + 4t9 + 6t10 + 2t11 + 2t12 + t16) / (1 – t2)3(1 – t3)4(1 – t5)2 =
1 + 3t2 + 4t3 + 8t4 + 16t5 + 32t6 + 48t7 + 95t8 + 152t9 + 246t10 + 398t11 + 622t12 + 926t13 + 1413t14 + 2054t15 + 2954t16 + 4214t17 + 5899t18 + 8110t19 + 11127t20 + ...

### V4+V4+V2

(1 + t3)(1 – 2t2 + 2t3 + 7t4 + 2t5 – 2t6 + t8) / (1 – t2)6(1 – t3)4 =
1 + 4t2 + 7t3 + 16t4 + 34t5 + 78t6 + 134t7 + 276t8 + 482t9 + 850t10 + 1430t11 + 2390t12 + 3772t13 + 5996t14 + 9150t15 + 13823t16 + 20418t17 + 29838t18 + 42643t19 + 60534t20 + ...

### V4+V4+V3

(1 + t2 + 4t4 + 8t5 + 21t6 + 32t7 + 54t8 + 74t9 + 102t10 + 124t11 + 154t12 + 176t13 + 200t14 + 206t15 + 200t16 + 176t17 + 154t18 + 124t19 + 102t20 + 74t21 + 54t22 + 32t23 + 21t24 + 8t25 + 4t26 + t28 + t30) / (1 – t2)2(1 – t3)4(1 – t4)(1 – t5)2(1 – t7)2 =
1 + 3t2 + 4t3 + 10t4 + 22t5 + 49t6 + 96t7 + 197t8 + 372t9 + 682t10 + 1214t11 + 2098t12 + 3508t13 + 5753t14 + 9194t15 + 14391t16 + 22108t17 + 33373t18 + 49524t19 + 72454t20 + ...

### V4+V4+V4

(1 + 4t3 + 6t4 + 3t5 + 10t6 + 21t7 + 21t8 + 10t9 + 3t10 + 6t11 + 4t12 + t15) / (1 – t2)6(1 – t3)6 =
(1 – 2t + 2t2 + 3t3 – 2t4 – t5 + 13t6 – t7 – 2t8 + 3t9 + 2t10 – 2t11 + t12) / (1 – t)2(1 – t2)5(1 – t3)5 =
1 + 6t2 + 10t3 + 27t4 + 63t5 + 147t6 + 285t7 + 621t8 + 1175t9 + 2232t10 + 4080t11 + 7271t12 + 12465t13 + 21162t14 + 34734t15 + 56082t16 + 88731t17 + 137961t18 + 210567t19 + 317220t20 + ...

## Four forms

### V1+V1+V1+V1

(1 + t2) / (1 – t2)5 =
1 + 6t2 + 20t4 + 50t6 + 105t8 + 196t10 + 336t12 + 540t14 + 825t16 + 1210t18 + 1716t20 + ...

Multivariate: (1 – tuvw) / (1 – tu)(1 – tv)(1 – uv)(1 – tw)(1 – uw)(1 – vw)

### V2+V1+V1+V1

(1 + t2 + 3t3 + t4 + t6) / (1 – t2)3(1 – t3)3 =
1 + 4t2 + 6t3 + 10t4 + 21t5 + 35t6 + 48t7 + 85t8 + 118t9 + 166t10 + 241t11 + 329t12 + 427t13 + 589t14 + 752t15 + 961t16 + 1237t17 + 1551t18 + 1903t19 + 2387t20 + ...

Multivariate: (1 + tuw + tvw + uvw – t2uvw – tu2vw – tuv2w – t2u2v2w2) / (1 – tu)(1 – tv)(1 – uv)(1 – w2)(1 – t2w)(1 – u2w)(1 – v2w)

### V2+V2+V1+V1

(1 + t)(1 – t + t2 + 2t3 + t4 – t5 + t6) / (1 – t2)4(1 – t3)3 =
1 + 4t2 + 6t3 + 13t4 + 24t5 + 47t6 + 70t7 + 125t8 + 188t9 + 287t10 + 422t11 + 618t12 + 852t13 + 1203t14 + 1624t15 + 2182t16 + 2884t17 + 3786t18 + 4856t19 + 6246t20 + ...

Multivariate: (1 + tuv + tuw + t2vw + tuvw + u2vw – t3uvw – t2u2vw – tu3vw – t2u2v2w – t2u2vw2 – t3u3v2w2) / (1 – tu)(1 – v2)(1 – vw)(1 – w2)(1 – t2v)(1 – u2v)(1 – t2w)(1 – u2w)

### V2+V2+V2+V1

(1 + t2 + t3 + 4t4 + t5 + t6 + t8) / (1 – t2)5(1 – t3)3 =
1 + 6t2 + 4t3 + 24t4 + 24t5 + 80t6 + 92t7 + 225t8 + 279t9 + 560t10 + 714t11 + 1267t12 + 1624t13 + 2640t14 + 3380t15 + 5145t16 + 6543t17 + 9484t18 + 11948t19 + 16659t20 + ...

Multivariate: (1 + uvw + t2uv + t2uw + t2vw – t2u2vw – t2uv2w – t2uvw2 – t4uvw – t4u2v2w2) / (1 – u2)(1 – uv)(1 – v2)(1 – uw)(1 – vw)(1 – w2)(1 – t2u)(1 – t2v)(1 – t2w)

### V2+V2+V2+V2

(1+t2+4t3+t4+t6) / (1 – t2)9 =
(1 – 2t + 4t2 – 2t3 + t4) / (1 – t)2(1 – t2)7 =
1 + 10t2 + 4t3 + 55t4 + 36t5 + 220t6 + 180t7 + 714t8 + 660t9 + 1992t10 + 1980t11 + 4950t12 + 5148t13 + 11220t14 + 12012t15 + 23595t16 + 25740t17 + 46618t18 + 51480t19 + 87373t20 + ...

Multivariate: (1 + tuv + tuw + tvw + uvw – t2uvw – tu2vw – tuv2w – tuvw2 – t2u2v2w2) / (1 – t2)(1 – tu)(1 – u2)(1 – tv)(1 – uv)(1 – v2)(1 – tw)(1 – uw)(1 – vw)(1 – w2)

### V3+V1+V1+V1

(1 + 13t4 + t6 + 13t8 + t12) / (1 – t2)3(1 – t4)4 =
1 + 3t2 + 23t4 + 62t6 + 195t8 + 426t10 + 958t12 + 1801t14 + 3384t16 + 5727t18 + 9611t20 + ...

### V3+V2+V1+V1

(1 + t + t2 + 4t3 + 11t4 + 20t5 + 29t6 + 33t7 + 39t8 + 41t9 + 39t10 + 33t11 + 29t12 + 20t13 + 11t14 + 4t15 + t16 + t17 + t18) / (1 + t)(1 – t2)(1 – t3)3(1 – t4)3(1 – t5) =
1 + 2t2 + 6t3 + 13t4 + 22t5 + 48t6 + 81t7 + 149t8 + 234t9 + 384t10 + 588t11 + 914t12 + 1312t13 + 1937t14 + 2727t15 + 3850t16 + 5238t17 + 7183t18 + 9548t19 + 12743t20 + ...

### V3+V2+V2+V1

(1 + t)(1 – t + 2t2 + t3 + 7t4 + 6t5 + 15t6 + 13t7 + 18t8 + 15t9 + 18t10 + 13t11 + 15t12 + 6t13 + 7t14 + t15 + 2t16 – t17 + t18) / (1 – t2)2(1 – t3)3(1 – t4)2(1 – t5)2 =
1 + 3t2 + 6t3 + 15t4 + 30t5 + 65t6 + 120t7 + 222t8 + 386t9 + 654t10 + 1062t11 + 1695t12 + 2610t13 + 3957t14 + 5860t15 + 8526t16 + 12186t17 + 17178t18 + 23838t19 + 32688t20 + ...

### V3+V2+V2+V2

(1 + t + 3t2 + 4t3 + 10t4 + 19t5 + 41t6 + 67t7 + 94t8 + 104t9 + 113t10 + 110t11 + 113t12 + 104t13 + 94t14 + 67t15 + 41t16 + 19t17 + 10t18 + 4t19 + 3t20 + t21 + t22) / (1 + t)(1 – t2)3(1 – t3)3(1 – t4)(1 – t5)3 =
1 + 6t2 + 4t3 + 25t4 + 34t5 + 101t6 + 166t7 + 362t8 + 604t9 + 1143t10 + 1851t11 + 3198t12 + 4994t13 + 8063t14 + 12186t15 + 18686t16 + 27395t17 + 40342t18 + 57524t19 + 82008t20 + ...

### V3+V3+V1+V1

(1 – 2t2 + 22t4 – 3t6 + 52t8 – 3t10 + 22t12 – 2t14 + t16) / (1 – t2)4(1 – t4)5 =
1 + 2t2 + 29t4 + 95t6 + 390t8 + 1056t10 + 2882t12 + 6525t14 + 14373t16 + 28514t18 + 54857t20 + ...

### V3+V3+V2+V1

(1 + 2t + 2t2 + 5t3 + 19t4 + 46t5 + 89t6 + 150t7 + 237t8 + 338t9 + 444t10 + 535t11 + 608t12 + 635t13 + 608t14 + 535t15 + 444t16 + 338t17 + 237t18 + 150t19 + 89t20 + 46t21 + 19t22 + 5t23 + 2t24 + 2t25 + t26) / (1 + t)2(1 – t2)(1 – t3)3(1 – t4)4(1 – t5)2 =
1 + 2t2 + 6t3 + 18t4 + 30t5 + 76t6 + 145t7 + 301t8 + 511t9 + 955t10 + 1577t11 + 2716t12 + 4234t13 + 6851t14 + 10358t15 + 15955t16 + 23247t17 + 34438t18 + 48890t19 + 70079t20 + ...

### V3+V3+V2+V2

(1 + 2t2 + 3t3 + 11t4 + 23t5 + 45t6 + 81t7 + 117t8 + 166t9 + 214t10 + 259t11 + 294t12 + 302t13 + 294t14 + 259t15 + 214t16 + 166t17 + 117t18 + 81t19 + 45t20 + 23t21 + 11t22 + 3t23 + 2t24 + t26) / (1 – t2)2(1 – t3)3(1 – t4)3(1 – t5)3 =
1 + 4t2 + 6t3 + 21t4 + 44t5 + 104t6 + 220t7 + 442t8 + 854t9 + 1588t10 + 2832t11 + 4919t12 + 8258t13 + 13564t14 + 21714t15 + 34056t16 + 52356t17 + 79084t18 + 117498t19 + 171946t20 + ...

### V3+V3+V3+V1

(1 – 2t2 + 27t4 + 10t6 + 130t8 + 30t10 + 130t12 + 10t14 + 27t16 – 2t18 + t20) / (1 – t2)5(1 – t4)6 =
1 + 3t2 + 38t4 + 168t6 + 798t8 + 2724t10 + 8666t12 + 23556t14 + 59713t16 + 137839t18 + 300846t20 + ...

### V3+V3+V3+V2

(1 + 3t + 6t2 + 13t3 + 37t4 + 96t5 + 221t6 + 463t7 + 908t8 + 1626t9 + 2684t10 + 4090t11 + 5812t12 + 7689t13 + 9515t14 + 11035t15 + 12070t16 + 12428t17 + 12070t18 + 11035t19 + 9515t20 + 7689t21 + 5812t22 + 4090t23 + 2684t24 + 1626t25 + 908t26 + 463t27 + 221t28 + 96t29 + 37t30 + 13t31 + 6t32 + 3t33 + t34) / (1 + t)3(1 – t2)(1 – t3)3(1 – t4)5(1 – t5)3 =
1 + 4t2 + 6t3 + 28t4 + 45t5 + 138t6 + 264t7 + 637t8 + 1145t9 + 2416t10 + 4222t11 + 8065t12 + 13426t13 + 23819t14 + 38265t15 + 64046t16 + 99227t17 + 158507t18 + 238397t19 + 366375t20 + ...

### V3+V3+V3+V3

(1 + 28t4 + 50t6 + 281t8 + 260t10 + 540t12 + 260t14 + 281t16 + 50t18 + 28t20 + t24) / (1 – t2)6(1 – t4)7 =
1 + 6t2 + 56t4 + 316t6 + 1666t8 + 6902t10 + 25312t12 + 80732t14 + 234093t16 + 619318t18 + 1526280t20 + ...

## Five forms

### V1+V1+V1+V1+V1

(1 + 3t2 + t4) / (1 – t2)7 =
1 + 10t2 + 50t4 + 175t6 + 490t8 + 1176t10 + 2520t12 + 4950t14 + 9075t16 + 15730t18 + 26026t20 + ...

Multivariate: (1 – tuvw – tuvx – tuwx – tvwx – uvwx + t2uvwx + tu2vwx + tuv2wx + tuvw2x + tuvwx2 – t2u2v2w2x2) / (1 – tu)(1 – tv)(1 – uv)(1 – tw)(1 – uw)(1 – vw)(1 – tx)(1 – ux)(1 – vx)(1 – wx)

### V2+V1+V1+V1+V1

(1 + 3t2 + 7t3 + 5t4 + 6t5 + 14t6 + 6t7 + 5t8 + 7t9 + 3t10 + t12) / (1 – t2)4(1 – t3)3(1 – t6) =
(1 + t + 3t2 + 9t3 + 11t4 + 8t5 + 11t6 + 9t7 + 3t8 + t9 + t10) / (1 + t)(1 – t2)4(1 – t3)4 =
1 + 7t2 + 10t3 + 27t4 + 55t5 + 112t6 + 181t7 + 357t8 + 545t9 + 913t10 + 1400t11 + 2164t12 + 3115t13 + 4673t14 + 6481t15 + 9212t16 + 12588t17 + 17247t18 + 22867t19 + 30708t20 + ...

### V2+V2+V1+V1+V1

(1 + t2 + 8t3 + 7t4 + 2t5 + 7t6 + 8t7 + t8 + t10) / (1 – t2)5(1 – t3)4 =
(1 – t + t2)(1 – t + 2t2 + 5t3 + 2t4 – t5 + t6) / (1 – t)2(1 – t2)3(1 – t3)4 =
1 + 6t2 + 12t3 + 27t4 + 66t5 + 134t6 + 246t7 + 474t8 + 818t9 + 1374t10 + 2274t11 + 3606t12 + 5556t13 + 8472t14 + 12532t15 + 18219t16 + 26142t17 + 36816t18 + 51090t19 + 70185t20 + ...

### V2+V2+V2+V1+V1

(1 + t2 + 6t3 + 10t4 + 6t5 + 6t6 + 10t7 + 6t8 + t9 + t11) / (1 – t2)6(1 – t3)4 =
(1 – t + t2)(1 + t2 + 5t3 + 10t4 + 5t5 + t6 + t8) / (1 – t)(1 – t2)5(1 – t3)4 =
1 + 7t2 + 10t3 + 37t4 + 70t5 + 177t6 + 320t7 + 672t8 + 1175t9 + 2158t10 + 3605t11 + 6113t12 + 9715t13 + 15527t14 + 23721t15 + 36122t16 + 53322t17 + 78222t18 + 111982t19 + 159273t20 + ...

### V2+V2+V2+V2+V1

(1 + 3t2 + 4t3 + 12t4 + 8t5 + 12t6 + 8t7 + 12t8 + 4t9 + 3t10 + t12) / (1 – t2)7(1 – t3)4 =
1 + 10t2 + 8t3 + 61t4 + 76t5 + 290t6 + 420t7 + 1138t8 + 1736t9 + 3837t10 + 5900t11 + 11431t12 + 17384t13 + 30717t14 + 45868t15 + 75716t16 + 110736t17 + 173492t18 + 248468t19 + 373439t20 + ...

### V2+V2+V2+V2+V2

(1 + 3t2 + 10t3 + 6t4 + 6t5 + 10t6 + 3t7 + t9) / (1 – t2)12 =
(1 – 3t + 9t2 – 9t3 + 9t4 – 3t5 + t6) / (1 – t)3(1 – t2)9 =
1 + 15t2 + 10t3 + 120t4 + 126t5 + 680t6 + 855t7 + 3045t8 + 4145t9 + 11427t10 + 16080t11 + 37310t12 + 53040t13 + 108810t14 + 154427t15 + 288990t16 + 406965t17 + 709410t18 + 988260t19 + 1628328t20 + ...

## Six forms

### V1+V1+V1+V1+V1+V1

(1 + t2)(1 + 5t2 + t4) / (1 – t2)9 =
1 + 15t2 + 105t4 + 490t6 + 1764t8 + 5292t10 + 13860t12 + 32670t14 + 70785t16 + 143143t18 + 273273t20 + ...