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Triangle T(n, k) = (2*n+1)!! * 2^(floor((n-1)/2) + floor(k/2) + 1) * Beta(floor(n/2) + floor((k-1)/2) + 2, floor((n-1)/2) + floor(k/2) + 2), read by rows.
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%I #9 Mar 08 2022 03:47:06

%S 1,5,2,14,7,6,126,54,54,24,594,297,264,132,120,7722,3432,3432,1560,

%T 1560,720,51480,25740,23400,11700,10800,5400,5040,875160,397800,

%U 397800,183600,183600,85680,85680,40320,7558200,3779100,3488400,1744200,1627920,813960,766080,383040,362880

%N Triangle T(n, k) = (2*n+1)!! * 2^(floor((n-1)/2) + floor(k/2) + 1) * Beta(floor(n/2) + floor((k-1)/2) + 2, floor((n-1)/2) + floor(k/2) + 2), read by rows.

%H G. C. Greubel, <a href="/A158868/b158868.txt">Rows n = 1..50 of the triangle, flattened</a>

%F T(n, k) = (2*n+1)!! * 2^(floor((n-1)/2) + floor(k/2) + 1) * Beta(floor(n/2) + floor((k-1)/2) + 2, floor((n-1)/2) + floor(k/2) + 2).

%F T(n, n) = n!. - _G. C. Greubel_, Mar 07 2022

%e Triangle begins as:

%e 1;

%e 5, 2;

%e 14, 7, 6;

%e 126, 54, 54, 24;

%e 594, 297, 264, 132, 120;

%e 7722, 3432, 3432, 1560, 1560, 720;

%e 51480, 25740, 23400, 11700, 10800, 5400, 5040;

%e 875160, 397800, 397800, 183600, 183600, 85680, 85680, 40320;

%e 7558200, 3779100, 3488400, 1744200, 1627920, 813960, 766080, 383040, 362880;

%t T[n_, k_]:= (2*n+1)!!*2^(1+Floor[n/2] +Floor[(k-1)/2])*Beta[Floor[n/2] +Floor[(k- 1)/2] +2, Floor[(n-1)/2] +Floor[k/2] +2];

%t Table[T[n, k], {n,10}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Mar 07 2022 *)

%o (Sage)

%o def T(n,k): return (2*n+1).multifactorial(2)*2^(1+(n//2)+((k-1)//2))*beta(2+(n//2)+((k-1)//2), 2+((n-1)//2)+(k//2))

%o flatten([[T(n,k) for k in (1..n)] for n in (1..10)]) # _G. C. Greubel_, Mar 07 2022

%Y Cf. A158867.

%K nonn,tabl

%O 1,2

%A _Roger L. Bagula_, Mar 28 2009

%E Edited by _G. C. Greubel_, Mar 07 2022