cp's OEIS Frontend

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].

%I A361380 #30 Apr 05 2023 13:31:32
%S A361380 1,2,3,6,17,56,215,922,4305,21894,119539,696632,4314925,28237146,
%T A361380 194602079,1407456694,10649642837,84100177424,691474151187,
%U A361380 5907288773554,52340230286509,480153099982726,4553711640946919,44584683333637168,450075389309517849
%N 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].
%H A361380 Alois P. Heinz, <a href="/A361380/b361380.txt">Table of n, a(n) for n = 0..576</a>
%F A361380 a(n) = Sum_{i=0..n} i! * [x^i] exp(exp(x)-(n-i)*x-1).
%F A361380 a(n) = Sum_{0<=j<=i<=n} binomial(i,j)*(i-n)^(i-j)*Bell(j).
%F A361380 a(n) mod 2 = A059841(n).
%p A361380 a:= n-> add(add(binomial(i, j)*(i-n)^(i-j)*combinat[bell](j), j=0..i), i=0..n):
%p A361380 seq(a(n), n=0..25);
%p A361380 # second Maple program:
%p A361380 a:= n-> add(i!*coeff(series(exp(exp(x)-(n-i)*x-1), x, i+1), x, i), i=0..n):
%p A361380 seq(a(n), n=0..25);
%p A361380 # third Maple program:
%p A361380 b:= proc(n, m) option remember;
%p A361380      `if`(n=0, 1, b(n-1, m+1)+m*b(n-1, m))
%p A361380     end:
%p A361380 a:= n-> add(b(i, i-n), i=0..n):
%p A361380 seq(a(n), n=0..25);
%o A361380 (Python)
%o A361380 from math import comb
%o A361380 from sympy import bell
%o A361380 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 A361380 Antidiagonal sums of A361781.
%Y A361380 Cf. A000110, A059841, A290219, A347420.
%K A361380 nonn
%O A361380 0,2
%A A361380 _Alois P. Heinz_, Mar 09 2023