A348085 a(n) = [x^n] Product_{k=1..2*n} 1/(1 - (2*k-1) * x).
1, 4, 170, 13776, 1652442, 262842580, 52116296024, 12380577235040, 3427841258566890, 1083931844930932140, 385417972804020879450, 152219732613102667656000, 66113646914860527721527960, 31319437721634527178263452656
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..277
Programs
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PARI
a(n) = polcoef(1/prod(k=1, 2*n, 1-(2*k-1)*x+x*O(x^n)), n);
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PARI
a(n) = if(n==0, 1, -sum(k=0, 2*n-1, (-1)^k*(2*k+1)^(3*n-1)*binomial(2*n-1, k))/(2^(2*n-1)*(2*n-1)!));
Formula
a(n) = A039755(3*n-1,2*n-1) for n > 0.
a(n) = (-1/(2^(2*n-1) * (2*n-1)!)) * Sum_{k=0..2*n-1} (-1)^k * (2*k+1)^(3*n-1) * binomial(2*n-1,k) for n > 0.
a(n) ~ 3^(3*n - 1/2) * n^(n - 1/2) / (sqrt(2*Pi*(1-c)) * (3 - 2*c)^n * c^(2*n - 1/2) * exp(n)), where c = -LambertW(-3*exp(-3/2)/2) = 0.62578253420128292... - Vaclav Kotesovec, Oct 02 2021
From Seiichi Manyama, May 16 2025: (Start)
a(n) = Sum_{k=0..n} 2^k * binomial(3*n-1,k+2*n-1) * Stirling2(k+2*n-1,2*n-1) for n > 0.
a(n) = Sum_{k=0..n} (-2)^k * (4*n-1)^(n-k) * binomial(3*n-1,k+2*n-1) * Stirling2(k+2*n-1,2*n-1) for n > 0. (End)