%I #43 Aug 16 2020 08:52:31

%S 1,8,52,272,1266,5344,20992,77584,272727,917936,2975492,9328736,

%T 28391410,84122688,243265848,688008048,1906476351,5184024112,

%U 13851270944,36409640400,94255399886,240529147072,605574003464,1505340071744

%N G.f.: Product_{k>=1} 1/(1-x^k)^(8*k).

%C Previous name was: Number of plane partitions of n into parts of 8 kinds.

%C In general, if g.f. = Product_{k>=1} 1/(1-x^k)^(m*k) and m > 0, then a(n) ~ 2^(m/36 - 1/3) * exp(m/12 + 3 * 2^(-2/3) * m^(1/3) * zeta(3)^(1/3) * n^(2/3)) * (m*zeta(3))^(m/36 + 1/6) / (A^m * sqrt(3*Pi) * n^(m/36 + 2/3)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - _Vaclav Kotesovec_, Mar 01 2015

%H Vaclav Kotesovec, <a href="/A193427/b193427.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.

%F G.f.: Product_{k>=1} (1-x^k)^(-8*k).

%F a(n) ~ 2^(19/18) * zeta(3)^(7/18) * exp(2/3 + 3 * 2^(1/3) * zeta(3)^(1/3) * n^(2/3)) / (A^8 * sqrt(3*Pi) * n^(8/9)), 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(8*Sum_{k>=1} x^k/(k*(1 - x^k)^2)). - _Ilya Gutkovskiy_, May 29 2018

%F Euler transform of 8*k. - _Georg Fischer_, Aug 15 2020

%p a:= proc(n) option remember; `if`(n=0, 1, 8*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 ANS = Block[{kmax = 50},

%t Coefficient[

%t Series[Product[1/(1 - x^k)^(8 k), {k, 1, kmax}], {x, 0, kmax}], x,

%t Range[0, kmax]]]

%t (* Second program: *)

%t a[n_] := a[n] = If[n==0, 1, 8*Sum[a[n-j]*DivisorSigma[2, j], {j, 1, n}]/n]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Feb 07 2017, after _Alois P. Heinz_ *)

%o (PARI) Vec(prod(k=1,100\2,(1-x^k)^(-8*k),1+O(x^101))) \\ _Charles R Greathouse IV_, Aug 09 2011

%Y Cf. A000219 (m=1), A161870 (m=2), A255610 (m=3), A255611 (m=4), A255612 (m=5), A255613 (m=6), A255614 (m=7).

%Y Cf. A023007, A023003, A000712.

%Y Column k=8 of A255961.

%K nonn

%O 0,2

%A _Martin Y. Veillette_, Jul 28 2011

%E New name from _Vaclav Kotesovec_, Mar 12 2015