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

%I A262803 #19 Jun 12 2024 04:26:27
%S A262803 1,2,5,12,26,54,110,214,409,764,1400,2520,4475,7828,13532,23124,39102,
%T A262803 65472,108658,178786,291883,472984,761119,1216696,1932898,3052462,
%U A262803 4793464,7487122,11634771,17991760,27692230,42431778,64737414,98360742,148853817,224405254
%N A262803 Expansion of Product_{k>=1} (1 + x^k) / (1 - x^k)^k.
%C A262803 Convolution of A000219 and A000009.
%H A262803 Vaclav Kotesovec, <a href="/A262803/b262803.txt">Table of n, a(n) for n = 0..2000</a>
%H A262803 Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], 2015-2016.
%F A262803 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.
%F A262803 Euler transform of [ 2, 2, 4, 4, 6, 6, ...]. - _Michael Somos_, Oct 02 2015
%F A262803 G.f.: Product_{k>0} (1 - x^k)^-(k + (k mod 2)). - _Michael Somos_, Oct 02 2015
%F A262803 Convolution square of A003293. - _Michael Somos_, Oct 02 2015
%e A262803 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 + ...
%t A262803 nmax = 40; CoefficientList[Series[Product[(1+x^k)/(1-x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]
%o A262803 (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 */
%Y A262803 Cf. A000009, A000219, A003293, A262667.
%K A262803 nonn
%O A262803 0,2
%A A262803 _Vaclav Kotesovec_, Oct 02 2015