A361380 Sum over the j-th term of the (n-j)-th inverse binomial transform of the Bell numbers (A000110) for all j in [n].

1, 2, 3, 6, 17, 56, 215, 922, 4305, 21894, 119539, 696632, 4314925, 28237146, 194602079, 1407456694, 10649642837, 84100177424, 691474151187, 5907288773554, 52340230286509, 480153099982726, 4553711640946919, 44584683333637168, 450075389309517849

Offset: 0

Author

Alois P. Heinz, Mar 09 2023

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..576

Crossrefs

Antidiagonal sums of A361781.

Cf. A000110, A059841, A290219, A347420.

Programs

Maple

a:= n-> add(add(binomial(i, j)*(i-n)^(i-j)*combinat[bell](j), j=0..i), i=0..n):
seq(a(n), n=0..25);
# second Maple program:
a:= n-> add(i!*coeff(series(exp(exp(x)-(n-i)*x-1), x, i+1), x, i), i=0..n):
seq(a(n), n=0..25);
# third Maple program:
b:= proc(n, m) option remember;
     `if`(n=0, 1, b(n-1, m+1)+m*b(n-1, m))
    end:
a:= n-> add(b(i, i-n), i=0..n):
seq(a(n), n=0..25);

Python

from math import comb
from sympy import bell
def A361380(n): return sum(comb(i,j)*(i-n)**(i-j)*bell(j) for i in range(n+1) for j in range(i+1)) # Chai Wah Wu, Apr 05 2023

Formula

a(n) = Sum_{i=0..n} i! * [x^i] exp(exp(x)-(n-i)*x-1).

a(n) = Sum_{0<=j<=i<=n} binomial(i,j)*(i-n)^(i-j)*Bell(j).

a(n) mod 2 = A059841(n).