A294755 Expansion of Product_{k>=1} ((1 + x^(2*k - 1))/(1 - x^(2*k - 1)))^(k^2).

1, 2, 2, 10, 18, 36, 86, 150, 326, 608, 1164, 2230, 4046, 7632, 13622, 24868, 44222, 78304, 138312, 240138, 418648, 718292, 1233494, 2097350, 3552370, 5987642, 10026088, 16745600, 27779030, 45970868, 75650248, 124100970, 202720814, 329909400, 535132036

Offset: 0

Author

Vaclav Kotesovec, Nov 08 2017

Keywords

• nonn

Comments

Convolution of A294749 and A294750.

In general, if g.f. = Product_{k>=1} ((1 + x^(2*k-1))/(1 - x^(2*k-1)))^(c2*k^2 + c1*k + c0) and c2 > 0, then a(n) ~ exp(Pi*sqrt(2) * c2^(1/4) * n^(3/4) / 3 + 7*(c1+c2) * Zeta(3) * sqrt(n) / (2*sqrt(c2) * Pi^2) + (Pi*(4*c0 + 2*c1 + c2) / (8*sqrt(2) * c2^(1/4)) - 49*(c1+c2)^2 * Zeta(3)^2 / (2^(3/2) * c2^(5/4) * Pi^5)) * n^(1/4) - (7*c0 + 21*c1/4 + c2 + 7*c0*c1/c2 + 7*c1^2/(2*c2)) * Zeta(3) / (4*Pi^2) + 22411*(c1+c2)^3 * Zeta(3)^3 / (196 * c2^2 * Pi^8) - (c1+c2)/24) * A^((c1+c2)/2) * (n^((c1+c2)/96 - 5/8) / (2^(c0/2 + (11*c1 + 5*c2)/48 + 9/4) * Pi^((c1+c2)/24) * c2^((c1+c2)/96 - 1/8))), where A is the Glaisher-Kinkelin constant A074962.

Links

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

Crossrefs

Cf. A206622, A292037, A294749, A294750.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Product[((1+x^(2*k-1))/(1-x^(2*k-1)))^(k^2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(sqrt(2)*Pi * n^(3/4)/3 + 7*Zeta(3) * sqrt(n) / (2*Pi^2) + (Pi / (8*sqrt(2)) - 49*Zeta(3)^2 / (2^(3/2) * Pi^5)) * n^(1/4) + 22411*Zeta(3)^3 / (196*Pi^8) - Zeta(3)/(4*Pi^2) - 1/24) * sqrt(A) / (2^(113/48) * Pi^(1/24) * n^(59/96)), where A is the Glaisher-Kinkelin constant A074962.