A035295 Expansion of sum ( q^n / product( 1-q^k, k=1..3*n), n=0..inf ).
1, 1, 2, 4, 7, 11, 17, 26, 38, 54, 77, 107, 148, 201, 272, 363, 483, 635, 832, 1081, 1399, 1796, 2299, 2924, 3707, 4673, 5874, 7348, 9166, 11384, 14102, 17404, 21425, 26285, 32172, 39259, 47799, 58036, 70318, 84985, 102507, 123354, 148163, 177582, 212464, 253692, 302411
Offset: 0
Keywords
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
Programs
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Mathematica
nmax = 50; CoefficientList[Series[Sum[x^k/Product[1 - x^j, {j, 1, 3*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^(3*k))*(1-x^(3*k-1))*(1-x^(3*k-2))];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(1/3) * exp(Pi*sqrt(2*n/3)) / (2^(5/3) * 3^(7/6) * Pi^(2/3) * n^(2/3)). - Vaclav Kotesovec, Jun 16 2025
Extensions
More terms from Vaclav Kotesovec, Jun 16 2025