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

Author

Seiichi Manyama, Jan 04 2024

Keywords

• sign
• tabl

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

Columns k=0..5 give A182386, (-1)^(n-1) * A000240(n), A001477, A368716, A368717, A368718.

Main diagonal gives A368725.

Cf. A337085.

Programs

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.