This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A193427 #43 Aug 16 2020 08:52:31
%S A193427 1,8,52,272,1266,5344,20992,77584,272727,917936,2975492,9328736,
%T A193427 28391410,84122688,243265848,688008048,1906476351,5184024112,
%U A193427 13851270944,36409640400,94255399886,240529147072,605574003464,1505340071744
%N A193427 G.f.: Product_{k>=1} 1/(1-x^k)^(8*k).
%C A193427 Previous name was: Number of plane partitions of n into parts of 8 kinds.
%C A193427 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 A193427 Vaclav Kotesovec, <a href="/A193427/b193427.txt">Table of n, a(n) for n = 0..1000</a>
%H A193427 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 A193427 G.f.: Product_{k>=1} (1-x^k)^(-8*k).
%F A193427 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 A193427 G.f.: exp(8*Sum_{k>=1} x^k/(k*(1 - x^k)^2)). - _Ilya Gutkovskiy_, May 29 2018
%F A193427 Euler transform of 8*k. - _Georg Fischer_, Aug 15 2020
%p A193427 a:= proc(n) option remember; `if`(n=0, 1, 8*add(
%p A193427 a(n-j)*numtheory[sigma][2](j), j=1..n)/n)
%p A193427 end:
%p A193427 seq(a(n), n=0..30); # _Alois P. Heinz_, Mar 11 2015
%t A193427 ANS = Block[{kmax = 50},
%t A193427 Coefficient[
%t A193427 Series[Product[1/(1 - x^k)^(8 k), {k, 1, kmax}], {x, 0, kmax}], x,
%t A193427 Range[0, kmax]]]
%t A193427 (* Second program: *)
%t A193427 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 A193427 (PARI) Vec(prod(k=1,100\2,(1-x^k)^(-8*k),1+O(x^101))) \\ _Charles R Greathouse IV_, Aug 09 2011
%Y A193427 Cf. A000219 (m=1), A161870 (m=2), A255610 (m=3), A255611 (m=4), A255612 (m=5), A255613 (m=6), A255614 (m=7).
%Y A193427 Cf. A023007, A023003, A000712.
%Y A193427 Column k=8 of A255961.
%K A193427 nonn
%O A193427 0,2
%A A193427 _Martin Y. Veillette_, Jul 28 2011
%E A193427 New name from _Vaclav Kotesovec_, Mar 12 2015