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A174117 Triangle T(n, k) = (2*k/(k+1))*binomial(n-1, k)*binomial(n+1, k) with T(n, 0) = T(n, n) = 1, read by rows.

%I A174117 #8 Feb 11 2021 22:56:57
%S A174117 1,1,1,1,3,1,1,8,8,1,1,15,40,15,1,1,24,120,120,24,1,1,35,280,525,280,
%T A174117 35,1,1,48,560,1680,1680,560,48,1,1,63,1008,4410,7056,4410,1008,63,1,
%U A174117 1,80,1680,10080,23520,23520,10080,1680,80,1,1,99,2640,20790,66528,97020,66528,20790,2640,99,1
%N A174117 Triangle T(n, k) = (2*k/(k+1))*binomial(n-1, k)*binomial(n+1, k) with T(n, 0) = T(n, n) = 1, read by rows.
%H A174117 G. C. Greubel, <a href="/A174117/b174117.txt">Rows n = 0..100 of the triangle, flattened</a>
%F A174117 Let c(n) = Product_{j=2..n} (j^2 - 1) for n > 1 otherwise 1 then the number triangle is given by T(n, k) = c(n)/(c(k)*c(n-k)).
%F A174117 From _G. C. Greubel_, Feb 11 2021: (Start)
%F A174117 T(n, k) = (2*k/(k+1))*binomial(n-1, k)*binomial(n+1, k) with T(n, 0) = T(n, n) = 1.
%F A174117 T(n, k) = 2*((n+1)*(n-k)/(k+1))*A001263(n, k).
%F A174117 Sum_{k=0..n} T(n, k) = (2/(n+2))*( (n^2-1)*C_{n} + 1), where C_{n} are the Catalan numbers (A000108). (End)
%e A174117 Triangle begins as:
%e A174117   1;
%e A174117   1,  1;
%e A174117   1,  3,    1;
%e A174117   1,  8,    8,     1;
%e A174117   1, 15,   40,    15,     1;
%e A174117   1, 24,  120,   120,    24,     1;
%e A174117   1, 35,  280,   525,   280,    35,     1;
%e A174117   1, 48,  560,  1680,  1680,   560,    48,     1;
%e A174117   1, 63, 1008,  4410,  7056,  4410,  1008,    63,    1;
%e A174117   1, 80, 1680, 10080, 23520, 23520, 10080,  1680,   80,  1;
%e A174117   1, 99, 2640, 20790, 66528, 97020, 66528, 20790, 2640, 99, 1;
%t A174117 (* First program *)
%t A174117 c[n_]:= If[n<2, 1, Product[i^2 -1, {i,2,n}]];
%t A174117 T[n_, k_]:= c[n]/(c[k]*c[n-k]);
%t A174117 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten
%t A174117 (* Second program *)
%t A174117 T[n_, k_]:= If[k==0 || k==n, 1, (2*k/(k+1))*Binomial[n+1, k]*Binomial[n-1, k]];
%t A174117 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 11 2021 *)
%o A174117 (Sage)
%o A174117 def T(n,k): return 1 if (k==0 or k==n) else (2*k/(k+1))*binomial(n-1, k)*binomial(n+1, k)
%o A174117 flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 11 2021
%o A174117 (Magma)
%o A174117 T:= func< n,k | k eq 0 or k eq n select 1 else (2*k/(k+1))*Binomial(n-1, k)*Binomial(n+1, k) >;
%o A174117 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 11 2021
%Y A174117 Cf. A174116, A174119, A174124, A174125.
%Y A174117 Cf. A000108, A001263.
%K A174117 nonn,tabl
%O A174117 0,5
%A A174117 _Roger L. Bagula_, Mar 08 2010
%E A174117 Edited by _G. C. Greubel_, Feb 11 2021