A326856 E.g.f.: Product_{k>=1} (1 + x^(4*k-3) / (4*k-3)).

1, 1, 0, 0, 0, 24, 144, 0, 0, 40320, 403200, 0, 0, 479001600, 8643317760, 29059430400, 0, 20922789888000, 475108274995200, 1871463083212800, 0, 2432902008176640000, 76354225980899328000, 525098781423304704000, 0, 620448401733239439360000

Offset: 0

Author

Vaclav Kotesovec, Jul 27 2019

Keywords

• nonn

Comments

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.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..440
• Vaclav Kotesovec, Graph - the asymptotic ratio (40000 terms)

Crossrefs

Cf. A007838, A088994, A326780, A326855.

Programs

Mathematica

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]!

Formula

a(n) ~ n! / (sqrt(2*Pi) * exp(gamma/4) * n^(3/4)), where gamma is the Euler-Mascheroni constant A001620.