%I #26 Nov 14 2022 04:58:38

%S 1,1,1,1,8,1,1,36,36,1,1,120,540,120,1,1,330,4950,4950,330,1,1,792,

%T 32670,108900,32670,792,1,1,1716,169884,1557270,1557270,169884,1716,1,

%U 1,3432,736164,16195608,44537922,16195608,736164,3432,1

%N Triangle T(n,m) read by rows: T(n,m) = Product_{i=0..6} binomial(n+i,m)/binomial(m+i,m).

%C Triangle of generalized binomial coefficients (n,k)_7; cf. A342889. - _N. J. A. Sloane_, Apr 03 2021

%H Seiichi Manyama, <a href="/A142467/b142467.txt">Rows n = 0..139, 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,m) = A142465(n,m)*binomial(n+6,m)/binomial(m+6,m).

%F Sum_{k=0..n} T(n, k) = A005365(n).

%e Triangle begins as:

%e 1;

%e 1, 1;

%e 1, 8, 1;

%e 1, 36, 36, 1;

%e 1, 120, 540, 120, 1;

%e 1, 330, 4950, 4950, 330, 1;

%e 1, 792, 32670, 108900, 32670, 792, 1;

%e 1, 1716, 169884, 1557270, 1557270, 169884, 1716, 1;

%e 1, 3432, 736164, 16195608, 44537922, 16195608, 736164, 3432, 1;

%e 1, 6435, 2760615, 131589315, 868489479, 868489479, 131589315, 2760615, 6435, 1;

%t T[n_, k_]:= Product[Binomial[n+j, k]/Binomial[k+j, k], {j,0,6}];

%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Nov 13 2022 *)

%o (PARI) T(n, k) = prod(j=0, 6, binomial(n+j, k)/binomial(k+j, k)); \\ _Seiichi Manyama_, Apr 01 2021

%o (Magma) [(&*[Binomial(n+j,k)/Binomial(k+j,k): j in [0..6]]): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 13 2022

%o (SageMath)

%o def A142467(n,k): return product(binomial(n+j,k)/binomial(k+j,k) for j in (0..6))

%o flatten([[A142467(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Nov 13 2022

%Y Cf. A001263, A005365 (row sums), A056939, A056940, A056941.

%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,easy,tabl

%O 0,5

%A _Roger L. Bagula_, Sep 20 2008

%E Edited by the Associate Editors of the OEIS, May 17 2009