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%I #9 Mar 08 2022 03:47:37
%S 1,5,4,14,14,12,126,108,108,96,594,594,528,528,480,7722,6864,6864,
%T 6240,6240,5760,51480,51480,46800,46800,43200,43200,40320,875160,
%U 795600,795600,734400,734400,685440,685440,645120,7558200,7558200,6976800,6976800,6511680,6511680,6128640,6128640,5806080
%N Triangle T(n, k) = (2*n+1)!! * 2^(1 + floor(n/2) + floor(k/2) + floor((k-1)/2)) * 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="/A158867/b158867.txt">Rows n = 1..50 of the triangle, flattened</a>
%F T(n, k) = (2*n+1)!! * 2^(1 + floor(n/2) + floor(k/2) + floor((k-1)/2)) * Beta(floor(n/2) + floor((k-1)/2) + 2, floor((n-1)/2) + floor(k/2) + 2).
%F T(n, n) = A268363(n). - _G. C. Greubel_, Mar 08 2022
%e Triangle begins as:
%e 1;
%e 5, 4;
%e 14, 14, 12;
%e 126, 108, 108, 96;
%e 594, 594, 528, 528, 480;
%e 7722, 6864, 6864, 6240, 6240, 5760;
%e 51480, 51480, 46800, 46800, 43200, 43200, 40320;
%e 875160, 795600, 795600, 734400, 734400, 685440, 685440, 645120;
%e 7558200, 7558200, 6976800, 6976800, 6511680, 6511680, 6128640, 6128640, 5806080;
%t T[n_, k_]:= (2*n+1)!!*2^(1+Floor[n/2]+Floor[k/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 08 2022 *)
%o (Sage)
%o def A158867(n,k): return (2*n+1).multifactorial(2)*2^(1+(n//2)+(k//2)+((k-1)//2))*beta(2+(n//2)+((k-1)//2), 2+((n-1)//2)+(k//2))
%o flatten([[A158867(n,k) for k in (1..n)] for n in (1..10)]) # _G. C. Greubel_, Mar 08 2022
%Y Cf. A158868, A268363.
%K nonn,tabl
%O 1,2
%A _Roger L. Bagula_, Mar 28 2009
%E Edited by _G. C. Greubel_, Mar 08 2022
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