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%I #33 Apr 13 2023 12:06:20
%S 1,2,7,18,47,110,258,568,1237,2600,5380,10870,21652,42350,81778,
%T 155676,292964,544846,1003078,1828128,3301952,5911740,10499385,
%U 18502582,32371011,56240816,97073055,166497412,283870383,481212656,811287037,1360575284,2270274785,3769835178,6230705170,10251665550,16794445441
%N Convolution square of A000219.
%C Equals [1,2,3,...] * [1,0,4,0,10,0,20,...] * [1,0,0,6,0,0,21,...] * [1,0,0,0,8,0,0,0,36,...] * ... - _Gary W. Adamson_, Jul 06 2009
%C Number of pairs of planar partitions of u and v where u + v = n. - _Joerg Arndt_, Apr 22 2014
%H Seiichi Manyama, <a href="/A161870/b161870.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Vaclav Kotesovec)
%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], Sep 30 2015, p. 19.
%H Paul Martin, Eric C. Rowell, and Fiona Torzewska, <a href="https://arxiv.org/abs/2301.13831">Classification of charge-conserving loop braid representations</a>, arXiv:2301.13831 [math.QA], 2023.
%F G.f.: 1 / prod(k>=1, (1-x^k)^k )^2. - _Joerg Arndt_, Apr 22 2014
%F a(n) ~ Zeta(3)^(2/9) * exp(1/6 + 3*n^(2/3)*(Zeta(3)/2)^(1/3)) / (A^2 * 2^(1/18) * sqrt(3*Pi) * n^(13/18)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... . - _Vaclav Kotesovec_, Feb 27 2015
%F G.f.: exp(2*Sum_{k>=1} x^k/(k*(1 - x^k)^2)). - _Ilya Gutkovskiy_, May 29 2018
%p a:= proc(n) option remember; `if`(n=0, 1, 2*add(
%p a(n-j)*numtheory[sigma][2](j), j=1..n)/n)
%p end:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Mar 12 2015
%t nn = 36; CoefficientList[Series[Product[1/(1 - x^i)^(2 i), {i, 1, nn}] , {x, 0, nn}], x] (* _Geoffrey Critzer_, Nov 29 2014 *)
%o (PARI) N=66;x='x+O('x^N); Vec(1/prod(k=1,N,(1-x^k)^k)^2) \\ _Joerg Arndt_, Apr 22 2014
%Y Cf. A000219.
%Y Column k=2 of A255961.
%K nonn
%O 0,2
%A _Gary W. Adamson_, Jun 20 2009
%E Added more terms, _Joerg Arndt_, Apr 22 2014
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