A294654 Expansion of Product_{k>=1} 1/((1 - x^(2*k-1))^(k*(5*k-3)/2)*(1 - x^(2*k))^(k*(5*k+3)/2)).

1, 1, 5, 12, 35, 81, 208, 475, 1123, 2505, 5617, 12192, 26368, 55797, 117255, 242660, 498126, 1010515, 2033662, 4053214, 8017622, 15729219, 30643069, 59268267, 113898873, 217480476, 412813600, 779042099, 1462188257, 2729852845, 5070966794, 9373909586, 17247473718

Offset: 0

Author

Ilya Gutkovskiy, Nov 06 2017

Keywords

• nonn

Comments

Euler transform of the generalized heptagonal numbers (A085787).

Links

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

Crossrefs

Cf. A000294, A085787, A278769, A294591, A294621, A294655.

Programs

Mathematica

nmax = 32; CoefficientList[Series[Product[1/((1 - x^(2 k - 1))^(k (5 k - 3)/2) (1 - x^(2 k))^(k (5 k + 3)/2)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (5 d (d + 1)/8 + (-1)^d (2 d + 1)/16 - 1/16), {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)^A085787(k).

a(n) ~ exp(Pi * (2/3)^(5/4) * n^(3/4) + 5*Zeta(3) * sqrt(3*n) / (2^(3/2) * Pi^2) - (75*3^(1/4) * Zeta(3)^2 / (2^(13/4) * Pi^5) + Pi / (2^(17/4) * 3^(3/4))) * n^(1/4) + 375 * Zeta(3)^3 / (8*Pi^8) - 5*Zeta(3) / (64*Pi^2) + 1/12) * Pi^(1/12) / (A * 2^(11/6) * 3^(7/48) * n^(31/48)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 07 2017