A264437 a(n) = Bernoulli(n, 1)*Pochhammer(n+1, n).

1, 1, 2, 0, -56, 0, 15840, 0, -17297280, 0, 50791104000, 0, -327856732600320, 0, 4080179409546240000, 0, -89192941330901151744000, 0, 3193957788339335451033600000, 0, -177450861021098776794591068160000, 0, 14644425624059165645548485417369600000, 0

Offset

0,3

Links

Table of n, a(n) for n=0..23.

Formula

a(n) = CatalanNumber(n)*Sum_{k=0..n} Eulerian1(n, k)*k!*(n - k)!*(-1)^k. # Peter Luschny, Aug 13 2022

Maple

seq(pochhammer(n+1, n)*bernoulli(n, 1), n=0..23);
# For illustration:
e1 := proc(n, k) combinat:-eulerian1(n, k) end:
catalan := n -> binomial(2*n, n)/(n + 1):
a := n -> catalan(n)*add(e1(n, k)*k!*(n - k)!*(-1)^k, k = 0..n): # Peter Luschny, Aug 13 2022

Mathematica

a[n_] := BernoulliB[n, 1]*Pochhammer[n+1, n];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Nov 13 2023 *)

Prog

(Sage)
def A264437(n):
    return bernoulli_polynomial(1, n)*factorial(2*n)//factorial(n)
[A264437(n) for n in range(24)]
(PARI) a(n) = subst(bernpol(n), 'x, 1) *(2*n)!/n!; \\ Michel Marcus, Nov 13 2023

Crossrefs

Cf. A001813, A027641, A268432, A000108 (Catalan), A173018 (Eulerian first order).

Sequence in context: A012420 A140800 A012694 * A306061 A195209 A098276

Adjacent sequences: A264434 A264435 A264436 * A264438 A264439 A264440

Keyword

sign

Author

Peter Luschny, Feb 14 2016

Extensions

Name and data changed to comply with Bernoulli(n,1) by Peter Luschny, Aug 13 2022

Status

approved