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A341200 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} j^k * (n-j)! * binomial(n,j)^2.

%I A341200 #15 Feb 07 2021 00:42:19
%S A341200 1,0,2,0,1,7,0,1,6,34,0,1,8,39,209,0,1,12,63,292,1546,0,1,20,117,544,
%T A341200 2505,13327,0,1,36,243,1168,5225,24306,130922,0,1,68,549,2800,12525,
%U A341200 55656,263431,1441729,0,1,132,1323,7312,33425,145836,653023,3154824,17572114
%N A341200 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} j^k * (n-j)! * binomial(n,j)^2.
%F A341200 About e.g.f. of column k, see A105218 or A105219 comment.
%e A341200 Square array begins:
%e A341200      1,    0,    0,     0,     0,     0, ...
%e A341200      2,    1,    1,     1,     1,     1, ...
%e A341200      7,    6,    8,    12,    20,    36, ...
%e A341200     34,   39,   63,   117,   243,   549, ...
%e A341200    209,  292,  544,  1168,  2800,  7312, ...
%e A341200   1546, 2505, 5225, 12525, 33425, 97125, ...
%t A341200 T[n_, k_] := Sum[If[j == k == 0, 1, j^k] * (n - j)! * Binomial[n, j]^2, {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Feb 06 2021 *)
%o A341200 (PARI) T(n, k) = sum(j=0, n, j^k*(n-j)!*binomial(n, j)^2);
%Y A341200 Columns k=0..4 gives A002720, A103194, A105219, A105218, A341196.
%Y A341200 Main diagonal gives A341197.
%Y A341200 Cf. A289192.
%K A341200 nonn,tabl
%O A341200 0,3
%A A341200 _Seiichi Manyama_, Feb 06 2021