A342972 Triangle T(n,k) read by rows: T(n,k) = Product_{j=0..n-1} binomial(n+j,k)/binomial(k+j,k).

1, 1, 1, 1, 3, 1, 1, 10, 10, 1, 1, 35, 105, 35, 1, 1, 126, 1176, 1176, 126, 1, 1, 462, 13860, 41580, 13860, 462, 1, 1, 1716, 169884, 1557270, 1557270, 169884, 1716, 1, 1, 6435, 2147145, 61408347, 184225041, 61408347, 2147145, 6435, 1

Offset

0,5

Comments

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).

Links

Seiichi Manyama, Rows n = 0..50, flattened

Formula

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

Example

Triangle begins:
  1;
  1,    1;
  1,    3,       1;
  1,   10,      10,        1;
  1,   35,     105,       35,         1;
  1,  126,    1176,     1176,       126,        1;
  1,  462,   13860,    41580,     13860,      462,       1;
  1, 1716,  169884,  1557270,   1557270,   169884,    1716,    1;
  1, 6435, 2147145, 61408347, 184225041, 61408347, 2147145, 6435, 1;

Mathematica

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 *)

Prog

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

Crossrefs

Row sums gives A342967.

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.

Sequence in context: A176157 A176156 A172339 * A060540 A087647 A100265

Adjacent sequences: A342969 A342970 A342971 * A342973 A342974 A342975

Keyword

nonn,tabl

Author

Seiichi Manyama, Apr 01 2021

Status

approved