A361380 - OEIS

%I #30 Apr 05 2023 13:31:32

%S 1,2,3,6,17,56,215,922,4305,21894,119539,696632,4314925,28237146,

%T 194602079,1407456694,10649642837,84100177424,691474151187,

%U 5907288773554,52340230286509,480153099982726,4553711640946919,44584683333637168,450075389309517849

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

%H Alois P. Heinz, <a href="/A361380/b361380.txt">Table of n, a(n) for n = 0..576</a>

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

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

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

%p a:= n-> add(add(binomial(i, j)*(i-n)^(i-j)*combinat[bell](j), j=0..i), i=0..n):

%p seq(a(n), n=0..25);

%p # second Maple program:

%p a:= n-> add(i!*coeff(series(exp(exp(x)-(n-i)*x-1), x, i+1), x, i), i=0..n):

%p seq(a(n), n=0..25);

%p # third Maple program:

%p b:= proc(n, m) option remember;

%p      `if`(n=0, 1, b(n-1, m+1)+m*b(n-1, m))

%p     end:

%p a:= n-> add(b(i, i-n), i=0..n):

%p seq(a(n), n=0..25);

%o (Python)

%o from math import comb

%o from sympy import bell

%o 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

%Y Antidiagonal sums of A361781.

%Y Cf. A000110, A059841, A290219, A347420.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Mar 09 2023