A158868 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.

1, 5, 2, 14, 7, 6, 126, 54, 54, 24, 594, 297, 264, 132, 120, 7722, 3432, 3432, 1560, 1560, 720, 51480, 25740, 23400, 11700, 10800, 5400, 5040, 875160, 397800, 397800, 183600, 183600, 85680, 85680, 40320, 7558200, 3779100, 3488400, 1744200, 1627920, 813960, 766080, 383040, 362880

Offset: 1

Author

Roger L. Bagula, Mar 28 2009

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
        1;
        5,       2;
       14,       7,       6;
      126,      54,      54,      24;
      594,     297,     264,     132,     120;
     7722,    3432,    3432,    1560,    1560,    720;
    51480,   25740,   23400,   11700,   10800,   5400,   5040;
   875160,  397800,  397800,  183600,  183600,  85680,  85680,  40320;
  7558200, 3779100, 3488400, 1744200, 1627920, 813960, 766080, 383040, 362880;
		

Links

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

Crossrefs

Cf. A158867.

Programs

Mathematica

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];
Table[T[n, k], {n,10}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 07 2022 *)

Sage

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))
flatten([[T(n,k) for k in (1..n)] for n in (1..10)]) # G. C. Greubel, Mar 07 2022

Formula

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).

T(n, n) = n!. - G. C. Greubel, Mar 07 2022

Extensions

Edited by G. C. Greubel, Mar 07 2022