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%I #21 May 14 2021 22:40:38
%S 1,1,1,1,10,1,1,55,55,1,1,220,1210,220,1,1,715,15730,15730,715,1,1,
%T 2002,143143,572572,143143,2002,1,1,5005,1002001,13026013,13026013,
%U 1002001,5005,1,1,11440,5725720,208416208,677352676,208416208,5725720,11440,1
%N Triangle read by rows: T(n, k) = c(n, q)/(c(k, q)*c(n-k, q)), where c(n, q) = Product_{j=1..n} (j+q)!/(j-1)! and q = 8.
%C Triangle of generalized binomial coefficients (n,k)_9; cf. A342889. - _N. J. A. Sloane_, Apr 03 2021
%C Row sums are: {1, 2, 12, 112, 1652, 32892, 862864, 28066040, 1105659414, 51177188350, 2734044648194, ...}.
%C These sequences (q >= 2) are a generalization of A056939.
%H G. C. Greubel, <a href="/A174109/b174109.txt">Rows n = 0..100 of triangle, flattened</a>
%H Johann Cigler, <a href="https://arxiv.org/abs/2103.01652">Pascal triangle, Hoggatt matrices, and analogous constructions</a>, arXiv:2103.01652 [math.CO], 2021.
%H Johann Cigler, <a href="https://www.researchgate.net/publication/349376205_Some_observations_about_Hoggatt_triangles">Some observations about Hoggatt triangles</a>, Universität Wien (Austria, 2021).
%F T(n, k, q) = c(n, q)/(c(k, q)*c(n-k, q)) where c(n, q) = Product_{j=1..n} (j+q)!/(j-1)! and q=8.
%F T(n, k, q) = (G(q+2)*G(k+1)*G(n+q+2)*G(n-k+1))/(G(n+1)*G(k+q+2)*G(n-k+q+ 2)), where G(x) is the Barnes G-function (see A000178). - _G. C. Greubel_, Apr 13 2019
%e Triangle begins as:
%e 1.
%e 1, 1.
%e 1, 10, 1.
%e 1, 55, 55, 1.
%e 1, 220, 1210, 220, 1.
%e 1, 715, 15730, 15730, 715, 1.
%e 1, 2002, 143143, 572572, 143143, 2002, 1.
%e 1, 5005, 1002001, 13026013, 13026013, 1002001, 5005, 1.
%e 1, 11440, 5725720, 208416208, 677352676, 208416208, 5725720, 11440, 1.
%t c[n_, q_]:= Product[i+j, {j, 0, q}, {i, 1, n}];
%t T[n_, m_, q_] = c[n, q]/(c[m, q]*c[n - m, q]);
%t Table[T[n, k, 8], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by _G. C. Greubel_, Apr 13 2019 *)
%t T[n_, k_, q_]:= (BarnesG[n+q+2]*BarnesG[k+1]*BarnesG[n-k+1]*BarnesG[q+2] )/(BarnesG[n-k+q+2]*BarnesG[k+q+2]*BarnesG[n+1]);
%t Table[T[n, k, 8], {n, 0, 10}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Apr 13 2019 *)
%o (PARI) {c(m,q) = prod(j=1,m, (j+q)!/(j-1)!)};
%o for(n=0,10, for(k=0,n, print1(c(n, 8)/(c(k, 8)*c(n-k, 8)), ", "))) \\ _G. C. Greubel_, Apr 13 2019
%Y Cf. A056939 (q=2), A056940 (q=3), A056941 (q=4), A142465 (q=5), A142467 (q=6), A142468 (q=7), this sequence (q=8).
%Y Triangles of generalized binomial coefficients (n,k)_m (or generalized Pascal triangles) for m = 1,...,12: A007318 (Pascal), A001263, A056939, A056940, A056941, A142465, A142467, A142468, A174109, A342889, A342890, A342891.
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Mar 08 2010
%E Edited by _G. C. Greubel_, Apr 13 2019
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