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%I #10 Jul 29 2019 02:58:41
%S 1,1,0,0,0,24,144,0,0,40320,403200,0,0,479001600,8643317760,
%T 29059430400,0,20922789888000,475108274995200,1871463083212800,0,
%U 2432902008176640000,76354225980899328000,525098781423304704000,0,620448401733239439360000
%N E.g.f.: Product_{k>=1} (1 + x^(4*k-3) / (4*k-3)).
%C In general, if c > 0, mod(d,c) <> 0 and e.g.f. = Product_{k>=1} (1 + x^(c*k+d) / (c*k+d)), then a(n) ~ n! * Gamma(1 + d/c) / (c^(1/c) * exp(gamma/c) * Gamma(1/c) * Gamma(1 + (d+1)/c) * n^(1 - 1/c)), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function.
%H Vaclav Kotesovec, <a href="/A326856/b326856.txt">Table of n, a(n) for n = 0..440</a>
%H Vaclav Kotesovec, <a href="/A326856/a326856.jpg">Graph - the asymptotic ratio (40000 terms)</a>
%F a(n) ~ n! / (sqrt(2*Pi) * exp(gamma/4) * n^(3/4)), where gamma is the Euler-Mascheroni constant A001620.
%t nmax = 30; CoefficientList[Series[Product[(1+x^(4*k-3)/(4*k-3)), {k, 1, Floor[nmax/4]+1}], {x, 0, nmax}], x] * Range[0, nmax]!
%Y Cf. A007838, A088994, A326780, A326855.
%K nonn
%O 0,6
%A _Vaclav Kotesovec_, Jul 27 2019
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