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A174116 Triangle T(n, k) = (n/2)*binomial(n-1, k-1)*binomial(n-1, k) with T(n, 0) = T(n, n) = 1, read by rows.

%I A174116 #5 Feb 11 2021 22:57:14
%S A174116 1,1,1,1,1,1,1,3,3,1,1,6,18,6,1,1,10,60,60,10,1,1,15,150,300,150,15,1,
%T A174116 1,21,315,1050,1050,315,21,1,1,28,588,2940,4900,2940,588,28,1,1,36,
%U A174116 1008,7056,17640,17640,7056,1008,36,1,1,45,1620,15120,52920,79380,52920,15120,1620,45,1
%N A174116 Triangle T(n, k) = (n/2)*binomial(n-1, k-1)*binomial(n-1, k) with T(n, 0) = T(n, n) = 1, read by rows.
%H A174116 G. C. Greubel, <a href="/A174116/b174116.txt">Rows n = 0..100 of the triangle, flattened</a>
%F A174116 Let c(n) = Product_{j=2..n} binomial(j,2) for n > 1 otherwise 1 then the number triangle is given by T(n, k) = c(n)/(c(k)*c(n-k)).
%F A174116 From _G. C. Greubel_, Feb 11 2021: (Start)
%F A174116 T(n, k) = (n/2)*binomial(n-1, k-1)*binomial(n-1, k) with T(n, 0) = T(n, n) = 1.
%F A174116 T(n, k) = binomial(n-k+1, 2)*A001263(n, k) with T(n, 0) = T(n, n) = 1.
%F A174116 Sum_{k=0..n} T(n,k) = binomial(n, 2)*C_{n-1} + 2 - [n=0], where C_{n} are the Catalan numbers (A000108) and [] is the Iverson bracket. (End)
%e A174116 Triangle begins as:
%e A174116   1;
%e A174116   1,  1;
%e A174116   1,  1,    1;
%e A174116   1,  3,    3,     1;
%e A174116   1,  6,   18,     6,     1;
%e A174116   1, 10,   60,    60,    10,     1;
%e A174116   1, 15,  150,   300,   150,    15,     1;
%e A174116   1, 21,  315,  1050,  1050,   315,    21,     1;
%e A174116   1, 28,  588,  2940,  4900,  2940,   588,    28,    1;
%e A174116   1, 36, 1008,  7056, 17640, 17640,  7056,  1008,   36,  1;
%e A174116   1, 45, 1620, 15120, 52920, 79380, 52920, 15120, 1620, 45, 1;
%t A174116 (* First program *)
%t A174116 c[n_]:= If[n<2, 1, Product[Binomial[j,2], {j, 2, n}]];
%t A174116 T[n_, k_]:= c[n]/(c[k]*c[n-k]);
%t A174116 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten
%t A174116 (* Second program *)
%t A174116 T[n_, k_]:= If[k==0 || k==n, 1, (n/2)*Binomial[n-1, k-1]*Binomial[n-1, k]];
%t A174116 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 11 2021 *)
%o A174116 (Sage)
%o A174116 def T(n,k): return 1 if (k==0 or k==n) else (n/2)*binomial(n-1, k-1)*binomial(n-1, k)
%o A174116 flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 11 2021
%o A174116 (Magma)
%o A174116 T:= func< n,k | k eq 0 or k eq n select 1 else (n/2)*Binomial(n-1, k-1)*Binomial(n-1, k) >;
%o A174116 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 11 2021
%Y A174116 Cf. A174117, A174119, A174124, A174125.
%Y A174116 Cf. A000108, A001263.
%K A174116 nonn,tabl
%O A174116 0,8
%A A174116 _Roger L. Bagula_, Mar 08 2010
%E A174116 Edited by _G. C. Greubel_, Feb 11 2021