A383282 a(n) = Sum_{k=0..n} (2*k+1) * (-1/2)^(n+k) * (2*k)! * (n-k)! * binomial(n,k)^2.

1, 1, 5, 51, 906, 24690, 956790, 49993650, 3387124440, 288755250840, 30247310482200, 3818739956308200, 571858101118458000, 100218359688123877200, 20319306632495415745200, 4719164981053010642154000, 1244680987088062472732784000, 369981708267221405777101680000

Offset

0,3

Links

Table of n, a(n) for n=0..17.

Formula

a(n) = (-1)^n * (n!)^2 * Sum_{k=0..n} (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*a(n-1) + (n-1)^2/2 * a(n-2) ) for n > 1.

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

Prog

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

Crossrefs

Cf. A383281.

Sequence in context: A243242 A111340 A124559 * A368838 A348023 A172403

Adjacent sequences: A383279 A383280 A383281 * A383283 A383284 A383285

Keyword

nonn

Author

Seiichi Manyama, Apr 22 2025

Status

approved