A262803 Expansion of Product_{k>=1} (1 + x^k) / (1 - x^k)^k.

1, 2, 5, 12, 26, 54, 110, 214, 409, 764, 1400, 2520, 4475, 7828, 13532, 23124, 39102, 65472, 108658, 178786, 291883, 472984, 761119, 1216696, 1932898, 3052462, 4793464, 7487122, 11634771, 17991760, 27692230, 42431778, 64737414, 98360742, 148853817, 224405254

Offset: 0

Author

Vaclav Kotesovec, Oct 02 2015

Keywords

• nonn

Comments

Convolution of A000219 and A000009.

Examples

			G.f. = 1 + 2*x + 5*x^2 + 12*x^3 + 26*x^4 + 54*x^5 + 110*x^6 + 214*x^7 + 409*x^8 + ...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
• Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.

Crossrefs

Cf. A000009, A000219, A003293, A262667.

Programs

Mathematica

nmax = 40; CoefficientList[Series[Product[(1+x^k)/(1-x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]

PARI

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k)^-(k%2 + k), 1 + x * O(x^n)), n))}; /* Michael Somos, Oct 02 2015 */

Formula

a(n) ~ exp(1/12 - Pi^4/(1728*zeta(3)) + Pi^2 * n^(1/3)/(3*2^(7/3)*zeta(3)^(1/3)) + 3*zeta(3)^(1/3) * n^(2/3)/2^(2/3)) * zeta(3)^(7/36) / (A * sqrt(3*Pi) * 2^(29/36) * n^(25/36)), where zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

Euler transform of [ 2, 2, 4, 4, 6, 6, ...]. - Michael Somos, Oct 02 2015

G.f.: Product_{k>0} (1 - x^k)^-(k + (k mod 2)). - Michael Somos, Oct 02 2015

Convolution square of A003293. - Michael Somos, Oct 02 2015