A326755 - OEIS

%I #19 Jul 24 2019 06:48:53

%S 1,0,1,0,6,24,90,504,7560,18144,485352,4626720,32033232,516559680,

%T 9142044912,64700161344,1804378343040,29722011830784,308081755013760,

%U 8202581858225664,184073277074529024,2067986628774743040,75069447974837132544,1673053361596502645760

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

%C 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).

%H Vaclav Kotesovec, <a href="/A326755/b326755.txt">Table of n, a(n) for n = 0..448</a>

%H D. H. Lehmer, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa21/aa21123.pdf">On reciprocally weighted partitions</a>, Acta Arithmetica XXI (1972), 379-388 (Theorem 7 needs a correction).

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

%F 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.

%t 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]!

%Y Cf. A007841, A294506, A309319, A326756.

%K nonn

%O 0,5

%A _Vaclav Kotesovec_, Jul 23 2019