OFFSET
0,3
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
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