A262803 - OEIS

%I #19 Jun 12 2024 04:26:27

%S 1,2,5,12,26,54,110,214,409,764,1400,2520,4475,7828,13532,23124,39102,

%T 65472,108658,178786,291883,472984,761119,1216696,1932898,3052462,

%U 4793464,7487122,11634771,17991760,27692230,42431778,64737414,98360742,148853817,224405254

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

%C Convolution of A000219 and A000009.

%H Vaclav Kotesovec, <a href="/A262803/b262803.txt">Table of n, a(n) for n = 0..2000</a>

%H 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 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 Euler transform of [ 2, 2, 4, 4, 6, 6, ...]. - _Michael Somos_, Oct 02 2015

%F G.f.: Product_{k>0} (1 - x^k)^-(k + (k mod 2)). - _Michael Somos_, Oct 02 2015

%F Convolution square of A003293. - _Michael Somos_, Oct 02 2015

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

%o (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 Cf. A000009, A000219, A003293, A262667.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Oct 02 2015