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Triangle read by rows, T(n, k) = (n - k)! * k! / floor(k / 2)! ^ 2.
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%I #11 Jul 30 2023 17:43:16

%S 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,

%T 20,5040,720,240,144,36,60,20,140,40320,5040,1440,720,144,180,40,140,

%U 70,362880,40320,10080,4320,720,720,120,280,70,630

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

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

%C 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).

%H Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.

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

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

%e [0] 1;

%e [1] 1, 1;

%e [2] 2, 1, 2;

%e [3] 6, 2, 2, 6;

%e [4] 24, 6, 4, 6, 6;

%e [5] 120, 24, 12, 12, 6, 30;

%e [6] 720, 120, 48, 36, 12, 30, 20;

%e [7] 5040, 720, 240, 144, 36, 60, 20, 140;

%e [8] 40320, 5040, 1440, 720, 144, 180, 40, 140, 70;

%e [9] 362880, 40320, 10080, 4320, 720, 720, 120, 280, 70, 630;

%p T := (n, k) -> (n - k)!*k! / iquo(k,2)! ^ 2:

%p seq(seq(T(n, k), k = 0..n), n = 0..9);

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

%K nonn,tabl

%O 0,4

%A _Peter Luschny_, Nov 13 2021