A368724 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) = n! * Sum_{j=0..n} (-1)^(n-j) * j^k/j!.
1, 0, 0, 0, 1, 1, 0, 1, 0, -2, 0, 1, 2, 3, 9, 0, 1, 6, 3, -8, -44, 0, 1, 14, 9, 4, 45, 265, 0, 1, 30, 39, 28, 5, -264, -1854, 0, 1, 62, 153, 100, -15, 6, 1855, 14833, 0, 1, 126, 543, 412, 125, 306, 7, -14832, -133496, 0, 1, 254, 1809, 1924, 1065, 546, -1799, 8, 133497, 1334961
Offset: 0
Examples
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, ...
Links
- Eric Weisstein's World of Mathematics, Bell Polynomial.
- Wikipedia, Touchard polynomials
Crossrefs
Programs
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PARI
T(n,k) = n!*sum(j=0, n, (-1)^(n-j)*j^k/j!);
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.