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

1,2

Links

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

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

Example

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;

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 *)

Prog

(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

Crossrefs

Cf. A158868, A268363.

Sequence in context: A329162 A344817 A094414 * A107984 A133178 A154225

Adjacent sequences: A158864 A158865 A158866 * A158868 A158869 A158870

Keyword

nonn,tabl

Author

Roger L. Bagula, Mar 28 2009

Extensions

Edited by G. C. Greubel, Mar 08 2022

Status

approved