A355171 a(n) = Sum_{k=0..n} binomial(n, k + 1)*k!*(n + 1)!/(k + 2)!.

0, 1, 7, 50, 406, 3804, 41028, 506064, 7084656, 111690720, 1967421600, 38425449600, 825970435200, 19404363283200, 495012834489600, 13632039812966400, 403120633444300800, 12740557701389414400, 428546132879432601600, 15284163618598275072000, 576073025410937628672000

Offset: 0

Author

Peter Luschny, Jun 22 2022

Keywords

• nonn

Crossrefs

Cf. A066667, A001705, A103213.

Programs

Maple

a := n -> n*(n + 1)!*hypergeom([1, 1, 1 - n], [2, 3], -1) / 2;
seq(simplify(a(n)), n = 0..20);

Mathematica

a[n_] := n * (n + 1)! * HypergeometricPFQ[{1, 1, 1 - n}, {2, 3}, -1]/2; Array[a, 21, 0] (* Amiram Eldar, Jun 22 2022 *)

Python

from math import comb, factorial
def A355171(n):
    f = factorial(n+1)
    return sum(f*comb(n,k+1)//(k+2)//(k+1) for k in range(n+1)) # Chai Wah Wu, Jun 22 2022

Formula

a(n) = n*(n + 1)!*hypergeom([1, 1, 1 - n], [2, 3], -1) / 2.

a(n) = Sum_{k=0..n} (-1)^(k+1)*k!*A066667(n, k + 1).

E.g.f.: log((1 - x) / (1 - 2*x)) / (1 - x)^2. - Mélika Tebni, Jun 23 2022

a(n) ~ 2^(n+2) * (n-1)!. - Vaclav Kotesovec, Feb 17 2024