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%I A173424 #9 Apr 15 2023 08:21:51
%S A173424 1,1,1,3,1,3,15,3,3,15,105,15,9,15,105,945,105,45,45,105,945,10395,
%T A173424 945,315,225,315,945,10395,135135,10395,2835,1575,1575,2835,10395,
%U A173424 135135,2027025,135135,31185,14175,11025,14175,31185,135135,2027025,34459425
%N A173424 Triangle read by rows: T(n, k) = (2*n - 2*k)!*(2*k)!/(2^n*(n - k)!*k!).
%F A173424 T(n, k) = A006882(2*n-2*k-1) * A006882(2*k-1).
%F A173424 T(n, k) = A001147(n-k) * A001147(k).
%F A173424 From _Peter Luschny_, Apr 15 2023: (Start)
%F A173424 T(n, k) = (1/Pi) * 2^n * Gamma(k + 1/2) * Gamma(n - k + 1/2).
%F A173424 T(n, k) = (2*n-1)!! * binomial(n, k) / binomial(2*n, 2*k). (End)
%e A173424 Triangle T(n, k) starts:
%e A173424 [0] 1;
%e A173424 [1] 1, 1;
%e A173424 [2] 3, 1, 3;
%e A173424 [3] 15, 3, 3, 15;
%e A173424 [4] 105, 15, 9, 15, 105;
%e A173424 [5] 945, 105, 45, 45, 105, 945;
%e A173424 [6] 10395, 945, 315, 225, 315, 945, 10395;
%e A173424 [7] 135135, 10395, 2835, 1575, 1575, 2835, 10395, 135135;
%e A173424 [8] 2027025, 135135, 31185, 14175, 11025, 14175, 31185, 135135, 2027025;
%p A173424 T := (n, k) -> doublefactorial(2*n-1) * binomial(n, k) / binomial(2*n, 2*k):
%p A173424 for n from 0 to 8 do seq(T(n, k), k = 0..n) od; # _Peter Luschny_, Apr 15 2023
%t A173424 t[n_, k_] = (2*n - 2*k)!*(2*k)!/(2^n*(n - k)!*k!);
%t A173424 Table[Table[t[n, k], {k, 0, n}], {n, 0, 10}]; Flatten[%]
%Y A173424 Cf. A034430 (row sums), A006882, A001147.
%K A173424 nonn,tabl
%O A173424 0,4
%A A173424 _Roger L. Bagula_, Feb 18 2010
%E A173424 Formula added by the Assoc. Editors of the OEIS, Feb 24 2010