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

%I A370418 #10 Mar 05 2024 11:25:06
%S A370418 1,1,2,4,6,24,36,48,120,720,576,720,1440,5040,40320,14400,17280,30240,
%T A370418 80640,362880,3628800,518400,604800,967680,2177280,7257600,39916800,
%U A370418 479001600,25401600,29030400,43545600,87091200,239500800,958003200,6227020800,87178291200
%N A370418 Triangle read by rows. T(n, k) = (n - k)! * (n + k)!.
%F A370418 Sum_{k=0..n} (-1)^k*T(n, k) = n!^2 / 2 + (-1)^n * (2*n + 2)! / (2*n + 2)^2.
%e A370418 Triangle starts:
%e A370418 [0]      1;
%e A370418 [1]      1,      2;
%e A370418 [2]      4,      6,     24;
%e A370418 [3]     36,     48,    120,     720;
%e A370418 [4]    576,    720,   1440,    5040,   40320;
%e A370418 [5]  14400,  17280,  30240,   80640,  362880,  3628800;
%e A370418 [6] 518400, 604800, 967680, 2177280, 7257600, 39916800, 479001600;
%p A370418 T := (n, k) -> (n - k)! * (n + k)!:
%p A370418 seq(seq(T(n, k), k = 0..n), n = 0..7);
%t A370418 Table[(n - k)!*(n + k)!, {n, 0, 7}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Mar 05 2024 *)
%Y A370418 Cf. A010050 (main diagonal), A009445 (subdiagonal), A001044 (column 0), A175430 (column 1), A024420 (bisection is alternating sum).
%Y A370418 Cf. A119502, A143084.
%K A370418 nonn,tabl
%O A370418 0,3
%A A370418 _Peter Luschny_, Feb 27 2024