OFFSET
0,10
LINKS
Eric Weisstein's World of Mathematics, Bell Polynomial.
Wikipedia, Touchard polynomials
FORMULA
T(0,k) = 0^k and T(n,k) = n^k - n * T(n-1,k) for n>0.
E.g.f. of column k: B_k(x) * exp(x) / (1+x), where B_n(x) = Bell polynomials.
EXAMPLE
Square array begins:
1, 0, 0, 0, 0, 0, 0, ...
0, 1, 1, 1, 1, 1, 1, ...
1, 0, 2, 6, 14, 30, 62, ...
-2, 3, 3, 9, 39, 153, 543, ...
9, -8, 4, 28, 100, 412, 1924, ...
-44, 45, 5, -15, 125, 1065, 6005, ...
265, -264, 6, 306, 546, 1386, 10626, ...
PROG
(PARI) T(n, k) = n!*sum(j=0, n, (-1)^(n-j)*j^k/j!);
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Seiichi Manyama, Jan 04 2024
STATUS
approved