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A351790 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 A351790 #17 Feb 19 2022 14:17:22
%S A351790 1,1,1,1,1,2,1,1,4,6,1,1,6,21,24,1,1,8,42,148,120,1,1,10,69,392,1305,
%T A351790 720,1,1,12,102,780,4600,13806,5040,1,1,14,141,1336,11145,64752,
%U A351790 170401,40320,1,1,16,186,2084,22200,191178,1063216,2403640,362880
%N A351790 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 A351790 E.g.f. of column k: 1/(1 - x*exp(k*x)).
%F A351790 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 A351790 Square array begins:
%e A351790     1,    1,    1,     1,     1,     1, ...
%e A351790     1,    1,    1,     1,     1,     1, ...
%e A351790     2,    4,    6,     8,    10,    12, ...
%e A351790     6,   21,   42,    69,   102,   141, ...
%e A351790    24,  148,  392,   780,  1336,  2084, ...
%e A351790   120, 1305, 4600, 11145, 22200, 39145, ...
%t A351790 T[n_, k_] := n!*(1 + Sum[(k*(n - j))^j/j!, {j, 1, n}]); Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Feb 19 2022 *)
%o A351790 (PARI) T(n, k) = n!*sum(j=0, n, (k*(n-j))^j/j!);
%o A351790 (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 A351790 Columns k=0..4 give A000142, A006153, A336950, A336951, A336952.
%Y A351790 Main diagonal gives A235328.
%Y A351790 Cf. A351761, A351791.
%K A351790 nonn,tabl
%O A351790 0,6
%A A351790 _Seiichi Manyama_, Feb 19 2022