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A258789 a(n) = [x^n] Product_{k=1..n} 1/(x^(2*k)*(1-x^k)).

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%I A258789 #9 Jun 11 2015 06:25:35
%S A258789 1,1,5,27,169,1115,7760,55748,411498,3101490,23785645,185064559,
%T A258789 1457664666,11602828475,93205739436,754751603157,6155229065861,
%U A258789 50515624923790,416930705579538,3458726257239312,28825340825747729,241245120218823892,2026803168946440648
%N A258789 a(n) = [x^n] Product_{k=1..n} 1/(x^(2*k)*(1-x^k)).
%H A258789 Vaclav Kotesovec, <a href="/A258789/b258789.txt">Table of n, a(n) for n = 0..380</a>
%F A258789 a(n) ~ c * d^n / n^2, where d = A258268 = 9.15337019245412246194853029240135454007332720412184884968926320147613... = r^3/(r-1), where r is the root of the equation polylog(2, 1-r) + 3*log(r)^2/2 = 0, c = 0.8069142856822510276258439534144172057548... .
%p A258789 T:=proc(n, k) option remember; `if`(n=0 or k=1, 1, T(n, k-1) + `if`(n<k, 0, T(n-k, k))) end proc: seq(T(n*(n+2), n), n=0..30);
%t A258789 Table[SeriesCoefficient[1/Product[x^(2*k)*(1-x^k), {k, 1, n}], {x, 0, n}], {n, 0, 30}]
%t A258789 Table[SeriesCoefficient[1/Product[1-x^k, {k, 1, n}], {x, 0, n*(n+2)}], {n, 0, 30}]
%Y A258789 Cf. A258268, A258788, A258791, A258793, A258794.
%K A258789 nonn
%O A258789 0,3
%A A258789 _Vaclav Kotesovec_, Jun 10 2015