A383869 a(n) = [x^n] 1/Product_{k=0..n} (1 - (n+k)*x).
1, 3, 55, 1890, 95781, 6427575, 537306484, 53791898160, 6275077781973, 835898091070185, 125195263380478655, 20825548503275385870, 3809430011164368694260, 759987002381075483922180, 164221938436980055710082200, 38209754165858724861944820000, 9524153723280871205135022364485
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Keywords
Programs
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Mathematica
Table[SeriesCoefficient[1/Product[(1 - (n + k)*x), {k, 0, n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 17 2025 *) -
PARI
a(n) = sum(k=0, n, (-1)^(n-k)*(n+k)^(2*n)*binomial(n, k))/n!;
Formula
a(n) = (1/n!) * Sum_{k=0..n} (-1)^(n-k) * (n+k)^(2*n) * binomial(n,k).
a(n) = Sum_{k=0..n} n^(n-k) * binomial(2*n,n+k) * Stirling2(n+k,n).
a(n) = Sum_{k=0..n} (-1)^k * (2*n)^(n-k) * binomial(2*n,n+k) * Stirling2(n+k,n).
a(n) ~ (r-1)^((r-1)*n) * (1+r)^(2*n + 1) * exp(n) * n^(n - 1/2) / (sqrt(2*Pi*(1 + (4-r)*r)) * r^(r*n)), where r = 2.106565648173949260853515992430777519716829316322... is the root of the equation exp(2/(1+r)) = r/(r-1). - Vaclav Kotesovec, May 17 2025