%I #19 May 29 2018 16:08:08

%S 1,7,42,203,882,3486,12880,44885,149170,475587,1462993,4359474,

%T 12628091,35656446,98372109,265701212,703800790,1830960824,4684293222,

%U 11798774953,29288385021,71714795158,173351031721,413964243476,977243358574,2281942600035,5273570826594

%N G.f.: Product_{k>=1} 1/(1-x^k)^(7*k).

%H Vaclav Kotesovec, <a href="/A255614/b255614.txt">Table of n, a(n) for n = 0..1000</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], Sep 30 2015, p. 19.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PlanePartition.html">Plane Partition</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Plane_partition">Plane partition</a>

%F G.f.: Product_{k>=1} 1/(1-x^k)^(7*k).

%F a(n) ~ 7^(13/36) * Zeta(3)^(13/36) * exp(7/12 + 3 * 2^(-2/3) * 7^(1/3) * Zeta(3)^(1/3) * n^(2/3)) / (A^7 * 2^(5/36) * sqrt(3*Pi) * n^(31/36)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... . - _Vaclav Kotesovec_, Feb 28 2015

%F G.f.: exp(7*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, 7*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 11 2015

%t nmax=50; CoefficientList[Series[Product[1/(1-x^k)^(7*k),{k,1,nmax}],{x,0,nmax}],x]

%Y Cf. A000219, A161870, A255610, A255611, A255612, A255613, A193427.

%Y Column k=7 of A255961.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Feb 28 2015

%E New name from _Vaclav Kotesovec_, Mar 12 2015