A385089 G.f.: Sum_{k>=0} x^k * Product_{j=1..3*k} (1 + x^j)/(1 - x^j).

1, 1, 3, 7, 15, 27, 47, 79, 127, 199, 307, 465, 695, 1025, 1493, 2151, 3069, 4337, 6075, 8441, 11639, 15933, 21667, 29281, 39337, 52555, 69849, 92375, 121595, 159347, 207939, 270259, 349911, 451377, 580223, 743341, 949241, 1208415, 1533763, 1941111, 2449841, 3083637

Offset: 0

Author

Vaclav Kotesovec, Jun 17 2025

Keywords

• nonn

Links

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

Crossrefs

Cf. A035295, A385067.

Cf. A207641, A385088, A385090, A385091, A385092.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Sum[x^k*Product[(1+x^j)/(1-x^j), {j, 1, 3*k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 50; p = 1; q = 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]]]; q = Expand[q*(1 + x^(3*k))*(1 + x^(3*k - 1))*(1 + x^(3*k - 2))]; q = Take[q, Min[nmax + 1, Exponent[q, x] + 1, Length[q]]]; s += x^k*q/p;, {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x]

Formula

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