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

0,3

Links

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

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

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 *)

Crossrefs

Cf. A363067, A385068.

Sequence in context: A329398 A353150 A035300 * A230118 A105807 A209616

Adjacent sequences: A035293 A035294 A035295 * A035297 A035298 A035299

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Vaclav Kotesovec, Jun 16 2025

Status

approved