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A262877 Expansion of Product_{k>=1} 1/(1-x^(3*k-2))^k.

%I A262877 #17 Apr 22 2021 08:29:30
%S A262877 1,1,1,1,3,3,3,6,9,9,13,19,23,28,42,51,62,84,108,127,170,219,261,328,
%T A262877 427,512,632,807,987,1190,1504,1838,2214,2744,3374,4036,4950,6060,
%U A262877 7260,8793,10748,12853,15459,18766,22473,26834,32425,38768,46136,55376,66168
%N A262877 Expansion of Product_{k>=1} 1/(1-x^(3*k-2))^k.
%C A262877 a(n) is the number of partitions of n into parts 3*k-2 of k kinds (k>=1). - _Joerg Arndt_, Oct 06 2015
%H A262877 Vaclav Kotesovec, <a href="/A262877/b262877.txt">Table of n, a(n) for n = 0..10000</a>
%H A262877 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
%H A262877 Vaclav Kotesovec, <a href="/A262877/a262877.jpg">Graph - The asymptotic ratio (70000 terms)</a>
%F A262877 a(n) ~ Zeta(3)^(13/108) * exp(d2 - Pi^4 / (972*Zeta(3)) + Pi^2 * n^(1/3) / (2^(1/3) * 3^(7/3) * Zeta(3)^(1/3)) + 3^(1/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (2^(41/108) * 3^(20/27) * sqrt(Pi) * n^(67/108)), where d2 = A263031 = Integral_{x=0..infinity} 1/x*(exp(-x)/(1 - exp(-3*x))^2 - 1/(9*x^2) - 2/(9*x) - 5*exp(-x)/36) = -0.0145374291832840336050202945022620903605414975934644413815... . - _Vaclav Kotesovec_, Oct 08 2015
%p A262877 with(numtheory):
%p A262877 a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A262877       `if`(irem(d+3, 3, 'r')=1, r, 0), d=divisors(j))*a(n-j), j=1..n)/n)
%p A262877     end:
%p A262877 seq(a(n), n=0..60);  # _Alois P. Heinz_, Oct 05 2015
%t A262877 nmax=100; CoefficientList[Series[Product[1/(1-x^(3k-2))^k, {k, 1, nmax}], {x, 0, nmax}], x]
%t A262877 nmax=100; CoefficientList[Series[E^Sum[1/j*x^j/(1-x^(3j))^2,{j,1,nmax}],{x,0,nmax}],x]
%Y A262877 Cf. A000219, A035528, A255610, A262876, A262878, A262879, A262883, A262947.
%K A262877 nonn
%O A262877 0,5
%A A262877 _Vaclav Kotesovec_, Oct 04 2015