A342891 - OEIS

%I #20 Apr 03 2021 03:12:31

%S 1,1,1,1,13,1,1,91,91,1,1,455,3185,455,1,1,1820,63700,63700,1820,1,1,

%T 6188,866320,4331600,866320,6188,1,1,18564,8836464,176729280,

%U 176729280,8836464,18564,1,1,50388,71954064,4892876352,19571505408,4892876352,71954064,50388,1

%N Triangle read by rows: T(n,k) = generalized binomial coefficients (n,k)_12 (n >= 0, 0 <= k <= n).

%C For references, links, programs, etc., see earlier sequences in this series, especially A342889.

%H Seiichi Manyama, <a href="/A342891/b342891.txt">Rows n = 0..139, flattened</a>

%F The generalized binomial coefficient (n,k)_m = Product_{j=1..k} binomial(n+m-j,m)/binomial(j+m-1,m).

%e Triangle begins:

%e   [1],

%e   [1, 1],

%e   [1, 13, 1],

%e   [1, 91, 91, 1],

%e   [1, 455, 3185, 455, 1],

%e   [1, 1820, 63700, 63700, 1820, 1],

%e   [1, 6188, 866320, 4331600, 866320, 6188, 1],

%e   [1, 18564, 8836464, 176729280, 176729280, 8836464, 18564, 1],

%e ...

%o (PARI) f(n, k, m) = prod(j=1, k, binomial(n-j+m, m)/binomial(j-1+m, m));

%o T(n, k) = f(n, k, 12); \\ _Seiichi Manyama_, Apr 02 2021

%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 _N. J. A. Sloane_, Apr 01 2021