A342972 - OEIS

%I #34 Apr 03 2021 08:38:46

%S 1,1,1,1,3,1,1,10,10,1,1,35,105,35,1,1,126,1176,1176,126,1,1,462,

%T 13860,41580,13860,462,1,1,1716,169884,1557270,1557270,169884,1716,1,

%U 1,6435,2147145,61408347,184225041,61408347,2147145,6435,1

%N Triangle T(n,k) read by rows: T(n,k) = Product_{j=0..n-1} binomial(n+j,k)/binomial(k+j,k).

%C Triangle read by rows: T(n,k) = generalized binomial coefficients (n,k)_n where (n,k)_m is Product_{j=1..k} binomial(n-j+m,m)/binomial(j-1+m,m).

%H Seiichi Manyama, <a href="/A342972/b342972.txt">Rows n = 0..50, flattened</a>

%F T(n,k) = Product_{j=0..k-1} binomial(2*n-1,n+j)/binomial(2*n-1,j).

%e Triangle begins:

%e   1;

%e   1,    1;

%e   1,    3,       1;

%e   1,   10,      10,        1;

%e   1,   35,     105,       35,         1;

%e   1,  126,    1176,     1176,       126,        1;

%e   1,  462,   13860,    41580,     13860,      462,       1;

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

%e   1, 6435, 2147145, 61408347, 184225041, 61408347, 2147145, 6435, 1;

%t T[n_, k_] := Product[Binomial[n + i, k]/Binomial[k + i, k], {i, 0, n - 1}]; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Apr 01 2021 *)

%o (PARI) T(n, k) = prod(j=0, n-1, binomial(n+j, k)/binomial(k+j, k));

%o (PARI) T(n, k) = prod(j=0, k-1, binomial(2*n-1, n+j)/binomial(2*n-1, j));

%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, n);

%Y Row sums gives A342967.

%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 _Seiichi Manyama_, Apr 01 2021