A351791 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))^j/j!.

1, 1, 1, 1, 1, 2, 1, 1, 0, 6, 1, 1, -2, -3, 24, 1, 1, -4, -6, -4, 120, 1, 1, -6, -3, 40, 25, 720, 1, 1, -8, 6, 132, 120, 114, 5040, 1, 1, -10, 21, 248, -375, -1872, -287, 40320, 1, 1, -12, 42, 364, -2120, -8298, -3920, -4152, 362880, 1, 1, -14, 69, 456, -5655, -12144, 86121, 155776, -1647, 3628800

Offset: 0

Author

Seiichi Manyama, Feb 19 2022

Keywords

• sign
• tabl

Examples

			Square array begins:
    1,  1,   1,    1,     1,     1, ...
    1,  1,   1,    1,     1,     1, ...
    2,  0,  -2,   -4,    -6,    -8, ...
    6, -3,  -6,   -3,     6,    21, ...
   24, -4,  40,  132,   248,   364, ...
  120, 25, 120, -375, -2120, -5655, ...
		

Crossrefs

Columns k=0..4 give A000142, (-1)^n * A302397(n), A336959, A351792, A351793.

Main diagonal gives (-1)^n * A302398(n).

Cf. A351776, A351790.

Programs

Mathematica

T[n_, k_] := n!*(1 + Sum[(-k*(n - j))^j/j!, {j, 1, n}]); Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 19 2022 *)

PARI

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

PARI

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

Formula

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

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