A370418 Triangle read by rows. T(n, k) = (n - k)! * (n + k)!.

1, 1, 2, 4, 6, 24, 36, 48, 120, 720, 576, 720, 1440, 5040, 40320, 14400, 17280, 30240, 80640, 362880, 3628800, 518400, 604800, 967680, 2177280, 7257600, 39916800, 479001600, 25401600, 29030400, 43545600, 87091200, 239500800, 958003200, 6227020800, 87178291200

Offset: 0

Author

Peter Luschny, Feb 27 2024

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[0]      1;
[1]      1,      2;
[2]      4,      6,     24;
[3]     36,     48,    120,     720;
[4]    576,    720,   1440,    5040,   40320;
[5]  14400,  17280,  30240,   80640,  362880,  3628800;
[6] 518400, 604800, 967680, 2177280, 7257600, 39916800, 479001600;
		

Crossrefs

Cf. A010050 (main diagonal), A009445 (subdiagonal), A001044 (column 0), A175430 (column 1), A024420 (bisection is alternating sum).

Cf. A119502, A143084.

Programs

Maple

T := (n, k) -> (n - k)! * (n + k)!:
seq(seq(T(n, k), k = 0..n), n = 0..7);

Mathematica

Table[(n - k)!*(n + k)!, {n, 0, 7}, {k, 0, n}] // Flatten (* Michael De Vlieger, Mar 05 2024 *)

Formula

Sum_{k=0..n} (-1)^k*T(n, k) = n!^2 / 2 + (-1)^n * (2*n + 2)! / (2*n + 2)^2.