A383281 a(n) = Sum_{k=0..n} (2*k+1) * (1/2)^(n+k) * (2*k)! * (n-k)! * binomial(n,k)^2.

1, 2, 11, 120, 2202, 61260, 2407770, 127116360, 8680455000, 744631438320, 78393873940200, 9938444069030400, 1493483322288157200, 262511581007832156000, 53360641241377862792400, 12420661873849173800856000, 3282370875452495120806512000, 977378127650967704776130016000

Offset: 0

Author

Seiichi Manyama, Apr 22 2025

Keywords

• nonn

Crossrefs

Cf. A002018.

Programs

PARI

a(n) = sum(k=0, n, (2*k+1)*(2*k)!*(n-k)!*binomial(n, k)^2/2^(n+k));

Formula

a(n) = (n!)^2 * Sum_{k=0..n} (-1)^k * (1/2)^(n-k) * binomial(-3/2,k)/(n-k)!.

a(n) = (n!)^2 * [x^n] 1/(1-x)^(3/2) * exp(x/2).

a(n) = n * ( (n+1)*a(n-1) - (n-1)^2/2 * a(n-2) ) for n > 1.

a(n) = A002018(n+1)/(n+1).

a(n) ~ 4 * sqrt(Pi) * n^(2*n + 3/2) / exp(2*n - 1/2). - Vaclav Kotesovec, Apr 24 2025