A385088 G.f.: Sum_{k>=0} x^k * Product_{j=1..2*k} (1 + x^j)/(1 - x^j).

1, 1, 3, 7, 13, 23, 39, 63, 101, 159, 243, 367, 547, 801, 1161, 1665, 2359, 3315, 4621, 6385, 8761, 11941, 16165, 21757, 29121, 38761, 51337, 67673, 88793, 116009, 150949, 195629, 252595, 324987, 416675, 532483, 678333, 861489, 1090913, 1377553, 1734761, 2178883

Offset: 0

Author

Vaclav Kotesovec, Jun 17 2025

Keywords

• nonn

Links

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

Crossrefs

Cf. A035294, A192433.

Cf. A207641, A385089, A385090, A385091, A385092.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Sum[x^k*Product[(1+x^j)/(1-x^j), {j, 1, 2*k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 50; p = 1; q = 1; s = 1; Do[p = Expand[p*(1 - x^(2*k))*(1 - x^(2*k - 1))]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; q = Expand[q*(1 + x^(2*k))*(1 + x^(2*k - 1))]; 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) ~ exp(Pi*sqrt(n)) / (16 * n^(3/4)).