OFFSET
0,9
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
Eric Weisstein's World of Mathematics, Bell Polynomial.
FORMULA
For k > 0, e.g.f. of column k: Sum_{j>=0} (exp(x)-1)^(k*j)/(k*j)!.
For k > 0, T(n,k) = ( Sum_{j=0..k-1} Bell_n(w^j) )/k, where Bell_n(x) is n-th Bell polynomial and w = exp(2*Pi*i/k).
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 0, 0, 0, 0, 0, ...
0, 2, 1, 0, 0, 0, 0, ...
0, 5, 3, 1, 0, 0, 0, ...
0, 15, 8, 6, 1, 0, 0, ...
0, 52, 25, 25, 10, 1, 0, ...
0, 203, 97, 91, 65, 15, 1, ...
PROG
(PARI) T(n, k) = sum(j=0, n, stirling(n, k*j, 2));
(PARI) T(n, k) = if(k==0, 0^n, n!*polcoef(sum(j=0, n\k, (exp(x+x*O(x^n))-1)^(k*j)/(k*j)!), n));
(PARI) Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
T(n, k) = if(k==0, 0^n, my(w=exp(2*Pi*I/k)); round(sum(j=0, k-1, Bell_poly(n, w^j)))/k);
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Oct 17 2022
STATUS
approved