A326860 E.g.f.: Product_{k>=1} (1 + x^(3*k-1)/(3*k-1)) / (1 - x^(3*k-1)/(3*k-1)).

1, 0, 2, 0, 12, 48, 180, 2016, 15120, 72576, 1424304, 11249280, 113164128, 2066238720, 22751977248, 303261573888, 6400216892160, 85934653249536, 1440131337066240, 34330891188013056, 549029461368181248, 11212163885207900160, 296439802585781976576

Offset: 0

Author

Vaclav Kotesovec, Jul 27 2019

Keywords

• nonn

Comments

In general, if c > 0, mod(d,c) <> 0, mod(d,c) <> 1 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) ~ Gamma(1 + (d-1)/c) * n^(2/c - 1) * n! / (c^(2/c) * exp(2*gamma/c) * Gamma(2/c) * Gamma(1 + (d+1)/c)), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function.

Crossrefs

Cf. A305199, A326755, A326857, A326859, A326861.

Programs

Mathematica

nmax = 30; CoefficientList[Series[Product[(1+x^(3*k-1)/(3*k-1))/(1-x^(3*k-1)/(3*k-1)), {k, 1, Floor[nmax/3]+1}], {x, 0, nmax}], x] * Range[0, nmax]!

Formula

a(n) ~ exp(-2*gamma/3) * Gamma(1/3)^2 * n! / (2 * 3^(1/6) * Pi * n^(1/3)), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function.