OFFSET
0,5
COMMENTS
Can be understood as a convolution matrix or as a sequence-to-triangle transformation similar to the partial Bell polynomials defined as: S -> T(n, k, S) = Sum_{j=0..n-k+1} C(n-1,j-1)*S(j)*T(n-j,k-1,S) if k != 0 else S(0)^n. Here S(n) = 1 for all n. The case S(n) = n gives the triangle of idempotent numbers A059297.
Conjecture: T(n,k) is the sum of two consecutive Stirling numbers of the second kind: T(n,k) = S2(n,k) + S2(n,k+1), see A008277. Checked up to n = 500. - Manfred Boergens, May 13 2024
EXAMPLE
Triangle starts:
1;
1, 1;
1, 2, 1;
1, 4, 4, 1;
1, 8, 13, 7, 1;
1, 16, 40, 35, 11, 1;
1, 32, 121, 155, 80, 16, 1;
1, 64, 364, 651, 490, 161, 22, 1;
The signed version is the inverse of A326326:
1;
-1, 1;
1, -2, 1;
-1, 4, -4, 1;
1, -8, 13, -7, 1;
-1, 16, -40, 35, -11, 1;
1, -32, 121, -155, 80, -16, 1;
-1, 64, -364, 651, -490, 161, -22, 1. - Peter Luschny, Jul 02 2019
MAPLE
# Implemented as a sequence transformation acting on f: n -> 1, 1, 1, 1, ... .
F := proc(n, k, f) option remember; `if`(k=0, f(0)^n,
add(binomial(n-1, j-1)*f(j)*F(n-j, k-1, f), j=0..n-k+1)) end:
for n from 0 to 7 do seq(F(n, k, j->1), k=0..n) od;
CROSSREFS
KEYWORD
AUTHOR
Peter Luschny, Apr 28 2015
STATUS
approved