A351761 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} k^(n-j) * (n-j)^j/j!.

1, 1, 0, 1, 1, 0, 1, 2, 4, 0, 1, 3, 12, 21, 0, 1, 4, 24, 102, 148, 0, 1, 5, 40, 279, 1160, 1305, 0, 1, 6, 60, 588, 4332, 16490, 13806, 0, 1, 7, 84, 1065, 11536, 84075, 281292, 170401, 0, 1, 8, 112, 1746, 25220, 282900, 1958058, 5598110, 2403640, 0

Offset: 0

Author

Seiichi Manyama, Feb 18 2022

Keywords

• nonn
• tabl

Examples

			Square array begins:
  1,    1,     1,     1,      1,      1, ...
  0,    1,     2,     3,      4,      5, ...
  0,    4,    12,    24,     40,     60, ...
  0,   21,   102,   279,    588,   1065, ...
  0,  148,  1160,  4332,  11536,  25220, ...
  0, 1305, 16490, 84075, 282900, 746525, ...
		

Crossrefs

Columns k=0..3 give A000007, A006153, A351762, A351763.

Main diagonal gives A351765.

Cf. A292860, A351776.

Programs

PARI

T(n, k) = n!*sum(j=0, n, k^(n-j)*(n-j)^j/j!);

PARI

T(n, k) = if(n==0, 1, k*n*sum(j=0, n-1, binomial(n-1, j)*T(j, k)));

Formula

E.g.f. of column k: 1/(1 - k*x*exp(x)).

T(0,k) = 1 and T(n,k) = k * n * Sum_{j=0..n-1} binomial(n-1,j) * T(j,k) for n > 0.