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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!.

%I A351791 #18 Feb 19 2022 14:17:34
%S A351791 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 A351791 720,1,1,-8,6,132,120,114,5040,1,1,-10,21,248,-375,-1872,-287,40320,1,
%U A351791 1,-12,42,364,-2120,-8298,-3920,-4152,362880,1,1,-14,69,456,-5655,-12144,86121,155776,-1647,3628800
%N 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!.
%F A351791 E.g.f. of column k: 1/(1 - x*exp(-k*x)).
%F A351791 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 A351791 Square array begins:
%e A351791     1,  1,   1,    1,     1,     1, ...
%e A351791     1,  1,   1,    1,     1,     1, ...
%e A351791     2,  0,  -2,   -4,    -6,    -8, ...
%e A351791     6, -3,  -6,   -3,     6,    21, ...
%e A351791    24, -4,  40,  132,   248,   364, ...
%e A351791   120, 25, 120, -375, -2120, -5655, ...
%t A351791 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 A351791 (PARI) T(n, k) = n!*sum(j=0, n, (-k*(n-j))^j/j!);
%o A351791 (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 A351791 Columns k=0..4 give A000142, (-1)^n * A302397(n), A336959, A351792, A351793.
%Y A351791 Main diagonal gives (-1)^n * A302398(n).
%Y A351791 Cf. A351776, A351790.
%K A351791 sign,tabl
%O A351791 0,6
%A A351791 _Seiichi Manyama_, Feb 19 2022