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

%I A349269 #11 Jul 30 2023 17:43:16
%S A349269 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 A349269 20,5040,720,240,144,36,60,20,140,40320,5040,1440,720,144,180,40,140,
%U A349269 70,362880,40320,10080,4320,720,720,120,280,70,630
%N A349269 Triangle read by rows, T(n, k) = (n - k)! * k! / floor(k / 2)! ^ 2.
%C A349269 Interpolates between the factorial numbers (A000142) and the swinging factorial numbers (A056040).
%C A349269 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 A349269 Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.
%F A349269 T(n, k) divides T(n, 0) for 0 <= k <= n.
%F A349269 Product_{k=0..n} T(n, k) is a square.
%e A349269 [0]      1;
%e A349269 [1]      1,     1;
%e A349269 [2]      2,     1,     2;
%e A349269 [3]      6,     2,     2,    6;
%e A349269 [4]     24,     6,     4,    6,   6;
%e A349269 [5]    120,    24,    12,   12,   6,  30;
%e A349269 [6]    720,   120,    48,   36,  12,  30,  20;
%e A349269 [7]   5040,   720,   240,  144,  36,  60,  20, 140;
%e A349269 [8]  40320,  5040,  1440,  720, 144, 180,  40, 140, 70;
%e A349269 [9] 362880, 40320, 10080, 4320, 720, 720, 120, 280, 70, 630;
%p A349269 T := (n, k) -> (n - k)!*k! / iquo(k,2)! ^ 2:
%p A349269 seq(seq(T(n, k), k = 0..n), n = 0..9);
%Y A349269 Cf. A349270 (row sums), A193282 (central coeffs.), A000142, A056040, A180064.
%K A349269 nonn,tabl
%O A349269 0,4
%A A349269 _Peter Luschny_, Nov 13 2021