A383280 a(n) = (3/2)^n * Sum_{k=0..n} (1/6)^k * (2*k)! * (n-k)! * binomial(n,k)^2.
1, 2, 9, 72, 954, 19980, 624510, 27420120, 1607036760, 120942324720, 11351106055800, 1298791163577600, 177888712528573200, 28728740092874421600, 5401708378739722249200, 1169716267087957140552000, 288993599402729842084464000, 80796133625685147464322528000
Offset: 0
Keywords
Programs
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PARI
a(n) = (3/2)^n*sum(k=0, n, (2*k)!*(n-k)!*binomial(n, k)^2/6^k);
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PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(serlaplace(1/sqrt(1-x) * exp(3*x/2)))) \\ Joerg Arndt, Apr 22 2025
Formula
a(n) = (n!)^2 * Sum_{k=0..n} (-1)^k * (3/2)^(n-k) * binomial(-1/2,k)/(n-k)!.
a(n) = (n!)^2 * [x^n] 1/sqrt(1-x) * exp(3*x/2).
a(n) = n * ( (n+1)*a(n-1) - 3*(n-1)^2/2 * a(n-2) ) for n > 1.
a(n) ~ 2 * sqrt(Pi) * n^(2*n + 1/2) / exp(2*n - 3/2). - Vaclav Kotesovec, Apr 24 2025