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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.
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%I #8 Feb 11 2021 22:56:57

%S 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 35,1,1,48,560,1680,1680,560,48,1,1,63,1008,4410,7056,4410,1008,63,1,

%U 1,80,1680,10080,23520,23520,10080,1680,80,1,1,99,2640,20790,66528,97020,66528,20790,2640,99,1

%N 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 G. C. Greubel, <a href="/A174117/b174117.txt">Rows n = 0..100 of the triangle, flattened</a>

%F 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 From _G. C. Greubel_, Feb 11 2021: (Start)

%F T(n, k) = (2*k/(k+1))*binomial(n-1, k)*binomial(n+1, k) with T(n, 0) = T(n, n) = 1.

%F T(n, k) = 2*((n+1)*(n-k)/(k+1))*A001263(n, k).

%F 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 Triangle begins as:

%e 1;

%e 1, 1;

%e 1, 3, 1;

%e 1, 8, 8, 1;

%e 1, 15, 40, 15, 1;

%e 1, 24, 120, 120, 24, 1;

%e 1, 35, 280, 525, 280, 35, 1;

%e 1, 48, 560, 1680, 1680, 560, 48, 1;

%e 1, 63, 1008, 4410, 7056, 4410, 1008, 63, 1;

%e 1, 80, 1680, 10080, 23520, 23520, 10080, 1680, 80, 1;

%e 1, 99, 2640, 20790, 66528, 97020, 66528, 20790, 2640, 99, 1;

%t (* First program *)

%t c[n_]:= If[n<2, 1, Product[i^2 -1, {i,2,n}]];

%t T[n_, k_]:= c[n]/(c[k]*c[n-k]);

%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten

%t (* Second program *)

%t T[n_, k_]:= If[k==0 || k==n, 1, (2*k/(k+1))*Binomial[n+1, k]*Binomial[n-1, k]];

%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 11 2021 *)

%o (Sage)

%o 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 flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 11 2021

%o (Magma)

%o 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 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 11 2021

%Y Cf. A174116, A174119, A174124, A174125.

%Y Cf. A000108, A001263.

%K nonn,tabl

%O 0,5

%A _Roger L. Bagula_, Mar 08 2010

%E Edited by _G. C. Greubel_, Feb 11 2021