A326861 E.g.f.: Product_{k>=1} (1 + x^(3*k-2)/(3*k-2)) / (1 - x^(3*k-2)/(3*k-2)).

1, 2, 4, 12, 60, 360, 2160, 16560, 149040, 1386720, 14592960, 174208320, 2173897440, 29413264320, 437473872000, 6792952636800, 112213292716800, 2002551280012800, 37194983281843200, 726119227314201600, 15112608758893324800, 326665495054151193600

Offset: 0

Author

Vaclav Kotesovec, Jul 27 2019

Keywords

• nonn

Comments

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.

Crossrefs

Cf. A305199, A326756, A326858, A326860.

Programs

Mathematica

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]!

Formula

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.