A035298 Expansion of sum ( q^n / product( 1-q^k, k=1..6*n), n=0..inf ).

1, 1, 2, 4, 7, 12, 19, 30, 44, 65, 93, 132, 183, 253, 343, 462, 616, 816, 1071, 1399, 1813, 2339, 2999, 3828, 4861, 6149, 7743, 9714, 12140, 15120, 18766, 23220, 28640, 35224, 43199, 52838, 64458, 78441, 95226, 115336, 139381, 168077, 202258, 242900, 291140, 348300, 415922

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

In general, for m>=1, if g.f. = Sum_{k>=0} x^k / Product_{j=1..m*k} (1 - x^j), then a(n) ~ Gamma(1/m) * exp(Pi*sqrt(2*n/3)) / (m * 2^((3*m + 1)/(2*m)) * 3^(1/(2*m)) * Pi^(1 - 1/m) * n^((m+1)/(2*m))). - Vaclav Kotesovec, Jun 17 2025

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A363068, A385070.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Sum[x^k/Product[1 - x^j, {j, 1, 6*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^(6*k))*(1-x^(6*k-1))*(1-x^(6*k-2))*(1-x^(6*k-3))*(1-x^(6*k-4))*(1-x^(6*k-5))];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(1/6) * exp(Pi*sqrt(2*n/3)) / (2^(31/12) * 3^(13/12) * Pi^(5/6) * n^(7/12)). - Vaclav Kotesovec, Jun 17 2025

Extensions

More terms from Vaclav Kotesovec, Jun 16 2025