A035297 Expansion of sum ( q^n / product( 1-q^k, k=1..5*n), n=0..inf ).

1, 1, 2, 4, 7, 12, 19, 29, 43, 63, 90, 127, 176, 241, 327, 439, 585, 773, 1015, 1322, 1714, 2208, 2831, 3610, 4585, 5794, 7297, 9149, 11433, 14233, 17665, 21846, 26943, 33123, 40614, 49656, 60565, 73671, 89414, 108254, 130785, 157649, 189654, 227671, 272802, 326236, 389446

Offset

0,3

Links

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

Formula

a(n) ~ Gamma(1/5) * exp(Pi*sqrt(2*n/3)) / (5 * 2^(8/5) * 3^(1/10) * Pi^(4/5) * n^(3/5)). - Vaclav Kotesovec, Jun 17 2025

Mathematica

nmax = 50; CoefficientList[Series[Sum[x^k/Product[1 - x^j, {j, 1, 5*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^(5*k))*(1-x^(5*k-1))*(1-x^(5*k-2))*(1-x^(5*k-3))*(1-x^(5*k-4))]; 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. A363068, A385069.

Sequence in context: A002622 A363276 A035301 * A272465 A153252 A079719

Adjacent sequences: A035294 A035295 A035296 * A035298 A035299 A035300

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Vaclav Kotesovec, Jun 16 2025

Status

approved