A326755 E.g.f.: Product_{k>=1} 1/(1 - x^(3*k-1)/(3*k-1)).

1, 0, 1, 0, 6, 24, 90, 504, 7560, 18144, 485352, 4626720, 32033232, 516559680, 9142044912, 64700161344, 1804378343040, 29722011830784, 308081755013760, 8202581858225664, 184073277074529024, 2067986628774743040, 75069447974837132544, 1673053361596502645760

Offset: 0

Author

Vaclav Kotesovec, Jul 23 2019

Keywords

• nonn

Comments

In the article by Lehmer, Theorem 7, p. 387, case b <> 0 and b <> 1, correct formula is W_n(S_a,b) ~ a^(-1/a) * exp(-gamma/a) * (Gamma((b-1)/a) / (Gamma(b/a) * Gamma(1/a))) * n^(1/a - 1).

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..448
• D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388 (Theorem 7 needs a correction).

Crossrefs

Cf. A007841, A294506, A309319, A326756.

Programs

Mathematica

nmax = 25; CoefficientList[Series[1/Product[(1-x^(3*k-1)/(3*k-1)), {k, 1, Floor[nmax/3] + 1}], {x, 0, nmax}], x] * Range[0, nmax]!

Formula

a(n) ~ 3^(1/6) * exp(-gamma/3) * Gamma(1/3) * n! / (2*Pi*n^(2/3)).

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