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A174125 Triangle T(n, k, q) = (q+1)*binomial(n, k)*(Pochhammer(q+1, n)/(Pochhammer(q+1, k)*Pochhammer(q+1, n-k))), with T(n, 0) = T(n, n) = 1, and q = 2, read by rows.

%I A174125 #7 Feb 11 2021 22:57:29
%S A174125 1,1,1,1,8,1,1,15,15,1,1,24,45,24,1,1,35,105,105,35,1,1,48,210,336,
%T A174125 210,48,1,1,63,378,882,882,378,63,1,1,80,630,2016,2940,2016,630,80,1,
%U A174125 1,99,990,4158,8316,8316,4158,990,99,1,1,120,1485,7920,20790,28512,20790,7920,1485,120,1
%N A174125 Triangle T(n, k, q) = (q+1)*binomial(n, k)*(Pochhammer(q+1, n)/(Pochhammer(q+1, k)*Pochhammer(q+1, n-k))), with T(n, 0) = T(n, n) = 1, and q = 2, read by rows.
%C A174125 Triangles of this class, depending upon q, are of the form T(n, k, q) = (q+1)*binomial(n, k)*(Pochhammer(q+1, n)/(Pochhammer(q+1, k)*Pochhammer(q+1, n-k))), with T(n, 0) = T(n, n) = 1, and have the row sums Sum_{k=0..n} T(n, k, q) = q*(q+1)*C_{n+q}/binomial(n+2*q, q-1) - 2*q + q*[n=0], where C_{n} are the Catalan numbers (A000108) and [] is the Iverson bracket. - _G. C. Greubel_, Feb 11 2021
%H A174125 G. C. Greubel, <a href="/A174125/b174125.txt">Rows n = 0..100 of the triangle, flattened</a>
%F A174125 Let c(n, q) = Product_{j=2..n} j*(j+q) for n > 2, otherwise 1, then the number triangle is given by T(n, k, q) = c(n, q)/(c(k, q)*c(n-k, q)) for q = 2.
%F A174125 From _G. C. Greubel_, Feb 11 2021: (Start)
%F A174125 T(n, k, q) = (q+1)*binomial(n, k)*(Pochhammer(q+1, n)/(Pochhammer(q+1, k)*Pochhammer(q+1, n-k))), with T(n, 0) = T(n, n) = 1, and q = 2.
%F A174125 Sum_{k=0..n} T(n, k, 1) = 6*A000108(n+2)/(n+4) - 4 + 2*[n=0]. (End)
%e A174125 Triangle begins as:
%e A174125   1;
%e A174125   1,   1;
%e A174125   1,   8,    1;
%e A174125   1,  15,   15,    1;
%e A174125   1,  24,   45,   24,     1;
%e A174125   1,  35,  105,  105,    35,     1;
%e A174125   1,  48,  210,  336,   210,    48,     1;
%e A174125   1,  63,  378,  882,   882,   378,    63,    1;
%e A174125   1,  80,  630, 2016,  2940,  2016,   630,   80,    1;
%e A174125   1,  99,  990, 4158,  8316,  8316,  4158,  990,   99,   1;
%e A174125   1, 120, 1485, 7920, 20790, 28512, 20790, 7920, 1485, 120, 1;
%t A174125 (* First program *)
%t A174125 c[n_, q_]:= If[n<2, 1, Product[i*(i+q), {i,2,n}]];
%t A174125 T[n_, m_, q_]:= c[n, q]/(c[k, q]*c[n-k, q]);
%t A174125 Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten
%t A174125 (* Second program *)
%t A174125 T[n_, k_, q_]:= If[k==0 || k==n, 1, (q+1)*Binomial[n, k]*(Pochhammer[q+1, n]/(Pochhammer[q+1, k]*Pochhammer[q+1, n-k]))];
%t A174125 Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 11 2021 *)
%o A174125 (Sage)
%o A174125 def T(n,k,q): return 1 if (k==0 or k==n) else (q+1)*binomial(n, k)*(rising_factorial(q+1, n)/(rising_factorial(q+1, k)*rising_factorial(q+1, n-k)))
%o A174125 flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 11 2021
%o A174125 (Magma)
%o A174125 c:= func< n,q | n lt 2 select 1 else (&*[j*(j+q): j in [2..n]]) >;
%o A174125 T:= func< n,k,q | c(n, q)/(c(k, q)*c(n-k, q)) >;
%o A174125 [T(n,k,2): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 11 2021
%Y A174125 Cf. A174124 (q=1), this sequence (q=2).
%Y A174125 Cf. A000108, A174116, A174117, A174119.
%K A174125 nonn,tabl
%O A174125 0,5
%A A174125 _Roger L. Bagula_, Mar 09 2010
%E A174125 Edited by _G. C. Greubel_, Feb 11 2021