A279932 - OEIS

%I #19 Mar 28 2018 03:58:43

%S 1,-5,5,0,30,-51,5,-130,220,-125,649,-605,870,-2695,1565,-4852,7915,

%T -6360,20625,-17880,33551,-61015,50865,-138510,135485,-224725,389025,

%U -359610,849525,-838970,1417404,-2195205,2275690,-4756040,4657940,-8315123,11174840,-13352315

%N Expansion of Product_{k>0} 1/(1 + x^k)^(k*5).

%C In general, if m >= 1 and g.f. = Product_{k>=1} 1/(1 + x^k)^(m*k), then a(n, m) ~ (-1)^n * exp(-m/12 + 3 * 2^(-5/3) * m^(1/3) * Zeta(3)^(1/3) * n^(2/3)) * 2^(m/18 - 5/6) * A^m * m^(1/6 - m/36) * Zeta(3)^(1/6 - m/36) * n^(m/36 - 2/3) / sqrt(3*Pi), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Apr 13 2017

%H Seiichi Manyama, <a href="/A279932/b279932.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) ~ (-1)^n * exp(-5/12 + 3 * 2^(-5/3) * (5*Zeta(3))^(1/3) * n^(2/3)) * A^5 * (5*Zeta(3))^(1/36) / (2^(5/9) * sqrt(3*Pi) * n^(19/36)), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Apr 13 2017

%F G.f.: exp(5*Sum_{k>=1} (-1)^k*x^k/(k*(1 - x^k)^2)). - _Ilya Gutkovskiy_, Mar 27 2018

%Y Column k=5 of A279928.

%Y Product_{k>0} 1/(1 + x^k)^(k*m): A027906 (m=-4), A255528 (m=1), A278710 (m=2), A279031 (m=3), A279411 (m=4), this sequence (m=5).

%K sign

%O 0,2

%A _Seiichi Manyama_, Apr 12 2017