A385068 G.f.: Sum_{k>=0} x^k * Product_{j=1..4*k} (1 + x^j).

1, 1, 2, 3, 5, 7, 9, 12, 16, 20, 26, 33, 41, 52, 65, 81, 99, 121, 147, 177, 214, 255, 304, 362, 429, 507, 596, 700, 820, 959, 1119, 1301, 1510, 1750, 2023, 2335, 2688, 3089, 3546, 4062, 4647, 5306, 6050, 6889, 7833, 8895, 10085, 11422, 12921, 14599, 16477, 18573, 20914

Offset: 0

Author

Vaclav Kotesovec, Jun 16 2025

Keywords

• nonn

Links

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

Crossrefs

Cf. A000009, A192433, A385067, A385069, A385070.

Cf. A035296.

Programs

Mathematica

nmax = 60; CoefficientList[Series[Sum[x^k*Product[1 + x^j, {j, 1, 4*k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 60; 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 += p*x^k;, {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x]

Formula

a(n) ~ Gamma(1/4) * 3^(1/8) * exp(Pi*sqrt(n/3)) / (2^(13/4) * Pi^(3/4) * n^(3/8)).