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

1,2

Links

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

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

Example

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;

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 *)

Prog

(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

Crossrefs

Cf. A158867.

Sequence in context: A286148 A369369 A194048 * A104634 A194008 A060422

Adjacent sequences: A158865 A158866 A158867 * A158869 A158870 A158871

Keyword

nonn,tabl

Author

Roger L. Bagula, Mar 28 2009

Extensions

Edited by G. C. Greubel, Mar 07 2022

Status

approved