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.

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, 2505, 13327, 0, 1, 36, 243, 1168, 5225, 24306, 130922, 0, 1, 68, 549, 2800, 12525, 55656, 263431, 1441729, 0, 1, 132, 1323, 7312, 33425, 145836, 653023, 3154824, 17572114

Offset: 0

Author

Seiichi Manyama, Feb 06 2021

Keywords

• nonn
• tabl

Examples

			Square array begins:
     1,    0,    0,     0,     0,     0, ...
     2,    1,    1,     1,     1,     1, ...
     7,    6,    8,    12,    20,    36, ...
    34,   39,   63,   117,   243,   549, ...
   209,  292,  544,  1168,  2800,  7312, ...
  1546, 2505, 5225, 12525, 33425, 97125, ...
		

Crossrefs

Columns k=0..4 gives A002720, A103194, A105219, A105218, A341196.

Main diagonal gives A341197.

Cf. A289192.

Programs

Mathematica

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 *)

PARI

T(n, k) = sum(j=0, n, j^k*(n-j)!*binomial(n, j)^2);

Formula

About e.g.f. of column k, see A105218 or A105219 comment.