%I #18 Feb 19 2022 14:17:34

%S 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,

%T 720,1,1,-8,6,132,120,114,5040,1,1,-10,21,248,-375,-1872,-287,40320,1,

%U 1,-12,42,364,-2120,-8298,-3920,-4152,362880,1,1,-14,69,456,-5655,-12144,86121,155776,-1647,3628800

%N 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!.

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

%F 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.

%e Square array begins:

%e 1, 1, 1, 1, 1, 1, ...

%e 1, 1, 1, 1, 1, 1, ...

%e 2, 0, -2, -4, -6, -8, ...

%e 6, -3, -6, -3, 6, 21, ...

%e 24, -4, 40, 132, 248, 364, ...

%e 120, 25, 120, -375, -2120, -5655, ...

%t 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 *)

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

%o (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)));

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

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

%Y Cf. A351776, A351790.

%K sign,tabl

%O 0,6

%A _Seiichi Manyama_, Feb 19 2022