A348087 a(n) = [x^n] Product_{k=1..n} 1/(1 - (2*k-1) * x).
1, 1, 13, 330, 12411, 618870, 38461522, 2863440580, 248440887123, 24616763946918, 2742625188929990, 339386813915985836, 46184075261030623710, 6854605372617955658940, 1101943692701420653738500, 190748265085183804327197000, 35373318817392757170821576835
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..317
Programs
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PARI
a(n) = polcoef(1/prod(k=1, 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, n-1, (-1)^k*(2*k+1)^(2*n-1)*binomial(n-1, k))/((-2)^(n-1)*(n-1)!));
Formula
a(n) = A039755(2*n-1,n-1) for n > 0.
a(n) = (1/((-2)^(n-1) * (n-1)!)) * Sum_{k=0..n-1} (-1)^k * (2*k+1)^(2*n-1) * binomial(n-1,k) for n > 0.
a(n) ~ 2^(3*n - 1) * n^(n - 1/2) / (sqrt(Pi*(1-c)) * (2-c)^n * c^(n - 1/2) * exp(n)), where c = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599... - Vaclav Kotesovec, Oct 02 2021
From Seiichi Manyama, May 16 2025: (Start)
a(n) = Sum_{k=0..n} 2^k * binomial(2*n-1,k+n-1) * Stirling2(k+n-1,n-1) for n > 0.
a(n) = Sum_{k=0..n} (-2)^k * (2*n-1)^(n-k) * binomial(2*n-1,k+n-1) * Stirling2(k+n-1,n-1) for n > 0. (End)