A158867 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.

1, 5, 4, 14, 14, 12, 126, 108, 108, 96, 594, 594, 528, 528, 480, 7722, 6864, 6864, 6240, 6240, 5760, 51480, 51480, 46800, 46800, 43200, 43200, 40320, 875160, 795600, 795600, 734400, 734400, 685440, 685440, 645120, 7558200, 7558200, 6976800, 6976800, 6511680, 6511680, 6128640, 6128640, 5806080

Offset: 1

Author

Roger L. Bagula, Mar 28 2009

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
        1;
        5,       4;
       14,      14,      12;
      126,     108,     108,      96;
      594,     594,     528,     528,     480;
     7722,    6864,    6864,    6240,    6240,    5760;
    51480,   51480,   46800,   46800,   43200,   43200,   40320;
   875160,  795600,  795600,  734400,  734400,  685440,  685440,  645120;
  7558200, 7558200, 6976800, 6976800, 6511680, 6511680, 6128640, 6128640, 5806080;
		

Links

• G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Crossrefs

Cf. A158868, A268363.

Programs

Mathematica

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];
Table[T[n, k], {n,10}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 08 2022 *)

Sage

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))
flatten([[A158867(n,k) for k in (1..n)] for n in (1..10)]) # G. C. Greubel, Mar 08 2022

Formula

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(n, n) = A268363(n). - G. C. Greubel, Mar 08 2022

Extensions

Edited by G. C. Greubel, Mar 08 2022