A385091 G.f.: Sum_{k>=0} x^k * Product_{j=1..5*k} (1 + x^j)/(1 - x^j).

1, 1, 3, 7, 15, 29, 53, 91, 151, 243, 381, 585, 881, 1305, 1907, 2753, 3931, 5559, 7793, 10835, 14955, 20501, 27921, 37801, 50889, 68139, 90777, 120353, 158827, 208683, 273037, 355791, 461839, 597273, 769661, 988411, 1265149, 1614215, 2053297, 2604113, 3293281, 4153407

Offset: 0

Author

Vaclav Kotesovec, Jun 17 2025

Keywords

• nonn

Links

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

Crossrefs

Cf. A035297, A385069.

Cf. A207641, A385088, A385089, A385090, A385092.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Sum[x^k*Product[(1+x^j)/(1-x^j), {j, 1, 5*k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 50; p = 1; q = 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]]]; q = Expand[q*(1 + x^(5*k))*(1 + x^(5*k - 1))*(1 + x^(5*k - 2))*(1 + x^(5*k - 3))*(1 + x^(5*k - 4))]; 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(1/5) * exp(Pi*sqrt(n)) / (5 * 2^(12/5) * Pi^(4/5) * n^(3/5)).