A173424 Triangle read by rows: T(n, k) = (2*n - 2*k)!*(2*k)!/(2^n*(n - k)!*k!).

1, 1, 1, 3, 1, 3, 15, 3, 3, 15, 105, 15, 9, 15, 105, 945, 105, 45, 45, 105, 945, 10395, 945, 315, 225, 315, 945, 10395, 135135, 10395, 2835, 1575, 1575, 2835, 10395, 135135, 2027025, 135135, 31185, 14175, 11025, 14175, 31185, 135135, 2027025, 34459425

Offset: 0

Author

Roger L. Bagula, Feb 18 2010

Keywords

• nonn
• tabl

Examples

			Triangle T(n, k) starts:
  [0]       1;
  [1]       1,      1;
  [2]       3,      1,     3;
  [3]      15,      3,     3,    15;
  [4]     105,     15,     9,    15,   105;
  [5]     945,    105,    45,    45,   105,   945;
  [6]   10395,    945,   315,   225,   315,   945, 10395;
  [7]  135135,  10395,  2835,  1575,  1575,  2835, 10395, 135135;
  [8] 2027025, 135135, 31185, 14175, 11025, 14175, 31185, 135135, 2027025;
		

Crossrefs

Cf. A034430 (row sums), A006882, A001147.

Programs

Maple

T := (n, k) -> doublefactorial(2*n-1) * binomial(n, k) / binomial(2*n, 2*k):
for n from 0 to 8 do seq(T(n, k), k = 0..n) od; # Peter Luschny, Apr 15 2023

Mathematica

t[n_, k_] = (2*n - 2*k)!*(2*k)!/(2^n*(n - k)!*k!);
Table[Table[t[n, k], {k, 0, n}], {n, 0, 10}]; Flatten[%]

Formula

T(n, k) = A006882(2*n-2*k-1) * A006882(2*k-1).

T(n, k) = A001147(n-k) * A001147(k).

From Peter Luschny, Apr 15 2023: (Start)

T(n, k) = (1/Pi) * 2^n * Gamma(k + 1/2) * Gamma(n - k + 1/2).

T(n, k) = (2*n-1)!! * binomial(n, k) / binomial(2*n, 2*k). (End)

Extensions

Formula added by the Assoc. Editors of the OEIS, Feb 24 2010