A302237 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(k*(k+1)/2).

1, 2, 8, 26, 76, 216, 590, 1554, 3988, 9988, 24464, 58794, 138866, 322808, 739658, 1672372, 3734848, 8245956, 18012114, 38952586, 83448832, 177194716, 373111970, 779430870, 1615995262, 3326484686, 6800794428, 13813260736, 27881653590, 55942340000, 111601021856

Offset: 0

Author

Ilya Gutkovskiy, Apr 03 2018

Keywords

• nonn

Comments

Convolution of the sequences A000294 and A028377.

Links

• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000217, A000294, A015128, A028377, A156616, A206622, A206623, A206624, A260916, A261386, A261452, A261519, A261520, A301554, A301555.

Programs

Mathematica

nmax = 30; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(k (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^A000217(k).

a(n) ~ exp(2*Pi*n^(3/4)/3 + 7*Zeta(3)*sqrt(n) / (2*Pi^2) - 49*Zeta(3)^2 * n^(1/4) / (4*Pi^5) + 22411 * Zeta(3)^3 / (392*Pi^8) - Zeta(3)/(8*Pi^2) + 1/24) * Pi^(1/24) / (sqrt(A) * 2^(25/12) * n^(61/96)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 08 2018

G.f.: A(x) = exp( 2*Sum_{n >= 0} x^(2*n+1)/((2*n+1)*(1 - x^(2*n+1))^3) ). Cf. A000122 and A156616. - Peter Bala, Dec 23 2021