A342890
Triangle read by rows: T(n,k) = generalized binomial coefficients (n,k)_11 (n >= 0, 0 <= k <= n).
Original entry on oeis.org
1, 1, 1, 1, 12, 1, 1, 78, 78, 1, 1, 364, 2366, 364, 1, 1, 1365, 41405, 41405, 1365, 1, 1, 4368, 496860, 2318680, 496860, 4368, 1, 1, 12376, 4504864, 78835120, 78835120, 4504864, 12376, 1, 1, 31824, 32821152, 1837984512, 6892441920, 1837984512, 32821152, 31824, 1
Offset: 0
Triangle begins:
[1],
[1, 1],
[1, 12, 1],
[1, 78, 78, 1],
[1, 364, 2366, 364, 1],
[1, 1365, 41405, 41405, 1365, 1],
[1, 4368, 496860, 2318680, 496860, 4368, 1],
[1, 12376, 4504864, 78835120, 78835120, 4504864, 12376, 1],
[1, 31824, 32821152, 1837984512, 6892441920, 1837984512, 32821152, 31824, 1],
...
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.
-
f(n, k, m) = prod(j=1, k, binomial(n-j+m, m)/binomial(j-1+m, m));
T(n, k) = f(n, k, 11); \\ Seiichi Manyama, Apr 02 2021
A342972
Triangle T(n,k) read by rows: T(n,k) = Product_{j=0..n-1} binomial(n+j,k)/binomial(k+j,k).
Original entry on oeis.org
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
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;
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.
-
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 *)
-
T(n, k) = prod(j=0, n-1, binomial(n+j, k)/binomial(k+j, k));
-
T(n, k) = prod(j=0, k-1, binomial(2*n-1, n+j)/binomial(2*n-1, j));
-
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);
Comments