A385090 G.f.: Sum_{k>=0} x^k * Product_{j=1..4*k} (1 + x^j)/(1 - x^j).

1, 1, 3, 7, 15, 29, 51, 87, 143, 227, 353, 537, 803, 1185, 1727, 2489, 3551, 5021, 7039, 9791, 13521, 18541, 25261, 34207, 46051, 61655, 82113, 108815, 143517, 188433, 246343, 320725, 415931, 537377, 691791, 887517, 1134863, 1446549, 1838235, 2329147, 2942849, 3708165

Offset: 0

Author

Vaclav Kotesovec, Jun 17 2025

Keywords

• nonn

Links

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

Crossrefs

Cf. A035296, A385068.

Cf. A207641, A385088, A385089, A385091, A385092.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Sum[x^k*Product[(1+x^j)/(1-x^j), {j, 1, 4*k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 50; p = 1; q = 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]]]; q = Expand[q*(1 + x^(4*k))*(1 + x^(4*k - 1))*(1 + x^(4*k - 2))*(1 + x^(4*k - 3))]; 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(1/4) * exp(Pi*sqrt(n)) / (2^(9/2) * Pi^(3/4) * n^(5/8)).