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%I #10 Jul 28 2019 03:17:50
%S 1,0,0,2,0,0,80,720,0,13440,172800,3628800,5913600,98841600,
%T 4420915200,92559667200,110702592000,6012444672000,234205087334400,
%U 6616915329024000,13708373852160000,771938716483584000,40374130262409216000,1172555787961958400000
%N E.g.f.: Product_{k>=1} 1/(1 - x^(4*k-1)/(4*k-1)).
%C In the article by Lehmer, Theorem 7, p. 387, case b <> 0 and b <> 1, correct formula is W_n(S_a,b) ~ a^(-1/a) * exp(-gamma/a) * (Gamma((b-1)/a) / (Gamma(b/a) * Gamma(1/a))) * n^(1/a - 1), where gamma is the Euler-Mascheroni constant (A001620) and Gamma() is the Gamma function.
%H Vaclav Kotesovec, <a href="/A326779/b326779.txt">Table of n, a(n) for n = 0..449</a>
%H Vaclav Kotesovec, <a href="/A326779/a326779.jpg">Graph - the asymptotic ratio (50000 terms)</a>
%H D. H. Lehmer, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa21/aa21123.pdf">On reciprocally weighted partitions</a>, Acta Arithmetica XXI (1972), 379-388 (Theorem 7 needs a correction).
%F a(n) ~ exp(-gamma/4) * n! / (2 * sqrt(Pi) * n^(3/4)), where gamma is the Euler-Mascheroni constant A001620.
%t nmax = 25; CoefficientList[Series[1/Product[(1-x^(4*k-1)/(4*k-1)), {k, 1, Floor[nmax/4] + 1}], {x, 0, nmax}], x] * Range[0, nmax]!
%Y Cf. A007841, A294506, A309319, A326755, A326756, A326780.
%K nonn
%O 0,4
%A _Vaclav Kotesovec_, Jul 24 2019
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