A294655 Expansion of Product_{k>=1} 1/((1 - x^(2*k-1))^(k*(3*k-2))*(1 - x^(2*k))^(k*(3*k+2))).

1, 1, 6, 14, 45, 106, 290, 683, 1698, 3918, 9179, 20640, 46444, 101819, 222092, 475886, 1012270, 2124725, 4425195, 9118705, 18648048, 37797126, 76062443, 151889787, 301296200, 593593192, 1162276735, 2261819285, 4376578818, 8421295585, 16118902083, 30694325652, 58164428059

Offset: 0

Author

Ilya Gutkovskiy, Nov 06 2017

Keywords

• nonn

Comments

Euler transform of the generalized octagonal numbers (A001082).

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
• N. J. A. Sloane, Transforms
• Eric Weisstein's World of Mathematics, Octagonal Number

Crossrefs

Cf. A000294, A001082, A274998, A294591, A294622, A294654, A294691, A294692.

Programs

Mathematica

nmax = 32; CoefficientList[Series[Product[1/((1 - x^(2 k - 1))^(k (3 k - 2)) (1 - x^(2 k))^(k (3 k + 2))), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (d^2 + d - Ceiling[d/2]^2), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 32}]

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^A001082(k+1).

a(n) ~ exp(Pi * 2^(3/2) * n^(3/4) / (3*5^(1/4)) + 3*Zeta(3) * sqrt(5*n) / (2*Pi^2) - (45*Zeta(3)^2 / Pi^5 + Pi/6) * 5^(1/4) * (n^(1/4) / 2^(5/2)) + 225 * Zeta(3)^3 / (4*Pi^8) - Zeta(3) / (32*Pi^2) + 1/8) * Pi^(1/8) / (A^(3/2) * 2^(77/48) * 5^(5/32) * n^(21/32)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 07 2017