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%I #6 Jul 27 2019 07:24:04
%S 1,2,4,12,60,360,2160,16560,149040,1386720,14592960,174208320,
%T 2173897440,29413264320,437473872000,6792952636800,112213292716800,
%U 2002551280012800,37194983281843200,726119227314201600,15112608758893324800,326665495054151193600
%N E.g.f.: Product_{k>=1} (1 + x^(3*k-2)/(3*k-2)) / (1 - x^(3*k-2)/(3*k-2)).
%C In general, if c > 0, d = 1-c and e.g.f. = Product_{k>=1} (1 + x^(c*k+d)/(c*k+d)) / (1 - x^(c*k+d)/(c*k+d)), then a(n) ~ 2 * n^(2/c) * n! / (c^(2/c) * exp(2*gamma/c) * Gamma(1 + 2/c)^2), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function.
%F a(n) ~ 3^(7/3) * exp(-2*gamma/3) * Gamma(1/3)^2 * n^(2/3) * n! / (8 * Pi^2), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function.
%t nmax = 30; CoefficientList[Series[Product[(1+x^(3*k-2)/(3*k-2))/(1-x^(3*k-2)/(3*k-2)), {k, 1, Floor[nmax/3]+1}], {x, 0, nmax}], x] * Range[0, nmax]!
%Y Cf. A305199, A326756, A326858, A326860.
%K nonn
%O 0,2
%A _Vaclav Kotesovec_, Jul 27 2019
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