A351776 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, 0, 0, 1, -3, 4, 3, 0, 1, -4, 12, -6, -4, 0, 1, -5, 24, -63, -8, -25, 0, 1, -6, 40, -204, 420, 150, 114, 0, 1, -7, 60, -465, 2288, -3435, -972, 287, 0, 1, -8, 84, -882, 7180, -32020, 33462, 3682, -4152, 0, 1, -9, 112, -1491, 17256, -138525, 537576, -379155, 6256, 1647, 0

Offset: 0

Author

Seiichi Manyama, Feb 19 2022

Keywords

• sign
• tabl

Examples

			Square array begins:
  1,   1,   1,     1,      1,       1, ...
  0,  -1,  -2,    -3,     -4,      -5, ...
  0,   0,   4,    12,     24,      40, ...
  0,   3,  -6,   -63,   -204,    -465, ...
  0,  -4,  -8,   420,   2288,    7180, ...
  0, -25, 150, -3435, -32020, -138525, ...
		

Crossrefs

Columns k=0..3 give A000007, A302397, A351777, A351778.

Main diagonal gives A351779.

Cf. A292861, A351761.

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.