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A355171
a(n) = Sum_{k=0..n} binomial(n, k + 1)*k!*(n + 1)!/(k + 2)!.
5
0, 1, 7, 50, 406, 3804, 41028, 506064, 7084656, 111690720, 1967421600, 38425449600, 825970435200, 19404363283200, 495012834489600, 13632039812966400, 403120633444300800, 12740557701389414400, 428546132879432601600, 15284163618598275072000, 576073025410937628672000
OFFSET
0,3
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Luschny, Jun 22 2022
STATUS
approved