A326779 E.g.f.: Product_{k>=1} 1/(1 - x^(4*k-1)/(4*k-1)).

1, 0, 0, 2, 0, 0, 80, 720, 0, 13440, 172800, 3628800, 5913600, 98841600, 4420915200, 92559667200, 110702592000, 6012444672000, 234205087334400, 6616915329024000, 13708373852160000, 771938716483584000, 40374130262409216000, 1172555787961958400000

Offset: 0

Author

Vaclav Kotesovec, Jul 24 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), 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..449
• Vaclav Kotesovec, Graph - the asymptotic ratio (50000 terms)
• D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388 (Theorem 7 needs a correction).

Crossrefs

Cf. A007841, A294506, A309319, A326755, A326756, A326780.

Programs

Mathematica

nmax = 25; CoefficientList[Series[1/Product[(1-x^(4*k-1)/(4*k-1)), {k, 1, Floor[nmax/4] + 1}], {x, 0, nmax}], x] * Range[0, nmax]!

Formula

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