A035296 Expansion of sum ( q^n / product( 1-q^k, k=1..4*n), n=0..inf ).

1, 1, 2, 4, 7, 12, 18, 28, 41, 60, 85, 119, 164, 225, 304, 408, 542, 716, 938, 1222, 1582, 2037, 2609, 3326, 4220, 5332, 6708, 8407, 10497, 13061, 16197, 20020, 24671, 30313, 37141, 45383, 55311, 67242, 81552, 98678, 119135, 143522, 172545, 207018, 247899, 296294, 353492

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A363067, A385068.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Sum[x^k/Product[1 - x^j, {j, 1, 4*k}], {k, 0, nmax}], {x, 0, nmax}], x]  (* Vaclav Kotesovec, Jun 16 2025 *)
nmax = 50; p=1; s=1; Do[p=Expand[p*(1-x^(4*k))*(1-x^(4*k-1))*(1-x^(4*k-2))*(1-x^(4*k-3))];p=Take[p, Min[nmax+1, Exponent[p, x]+1, Length[p]]];s+=x^k/p;, {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 16 2025 *)

Formula

a(n) ~ Gamma(1/4) * exp(Pi*sqrt(2*n/3)) / (2^(29/8) * 3^(1/8) * Pi^(3/4) * n^(5/8)). - Vaclav Kotesovec, Jun 17 2025

Extensions

More terms from Vaclav Kotesovec, Jun 16 2025