A349269 Triangle read by rows, T(n, k) = (n - k)! * k! / floor(k / 2)! ^ 2.

1, 1, 1, 2, 1, 2, 6, 2, 2, 6, 24, 6, 4, 6, 6, 120, 24, 12, 12, 6, 30, 720, 120, 48, 36, 12, 30, 20, 5040, 720, 240, 144, 36, 60, 20, 140, 40320, 5040, 1440, 720, 144, 180, 40, 140, 70, 362880, 40320, 10080, 4320, 720, 720, 120, 280, 70, 630

Offset: 0

Author

Peter Luschny, Nov 13 2021

Keywords

• nonn
• tabl

Comments

Interpolates between the factorial numbers (A000142) and the swinging factorial numbers (A056040).

The identity T(n, 0) = T(n, n)*T(floor(n/2), 0)^2 was investigated as a basis for an efficient implementation of the computation of the factorial numbers (see link).

Examples

			[0]      1;
[1]      1,     1;
[2]      2,     1,     2;
[3]      6,     2,     2,    6;
[4]     24,     6,     4,    6,   6;
[5]    120,    24,    12,   12,   6,  30;
[6]    720,   120,    48,   36,  12,  30,  20;
[7]   5040,   720,   240,  144,  36,  60,  20, 140;
[8]  40320,  5040,  1440,  720, 144, 180,  40, 140, 70;
[9] 362880, 40320, 10080, 4320, 720, 720, 120, 280, 70, 630;
		

Links

• Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.

Crossrefs

Cf. A349270 (row sums), A193282 (central coeffs.), A000142, A056040, A180064.

Programs

Maple

T := (n, k) -> (n - k)!*k! / iquo(k,2)! ^ 2:
seq(seq(T(n, k), k = 0..n), n = 0..9);

Formula

T(n, k) divides T(n, 0) for 0 <= k <= n.

Product_{k=0..n} T(n, k) is a square.