A120258 Triangle of central coefficients of generalized Pascal-Narayana triangles.

1, 1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 20, 20, 4, 1, 1, 70, 175, 50, 5, 1, 1, 252, 1764, 980, 105, 6, 1, 1, 924, 19404, 24696, 4116, 196, 7, 1, 1, 3432, 226512, 731808, 232848, 14112, 336, 8, 1, 1, 12870, 2760615, 24293412, 16818516, 1646568, 41580, 540, 9, 1

Offset

0,5

Comments

Columns are the central coefficients of the triangles T(n, k;r) with T(n, k;r)=Product{j=0..r, C(n+j, k+j)/C(n-k+j, j)}*[k<=n]; (r=0,A007318), (r=1;A001263),(r=2,A056939),(r=3,A056940),(r=4,A056941). Essentially A103905 as a number triangle with an extra diagonal of 1's. Central coefficients T(2n, n) are A008793. Row sums are A120259. Diagonal sums are A120260.

Links

Seiichi Manyama, Rows n = 0..100, flattened

Formula

Number triangle T(n, k)=[k<=n]*Product{j=0..k-1, C(2n-2k+j, n-k)/C(n-k+j, j)}

As a square array, this is T(n,m)=product{k=1..m, product{j=1..n, product{i=1..n, (i+j+k-1)/(i+j+k-2)}}}; - Paul Barry, May 13 2008

Example

Triangle begins:
  1;
  1,   1;
  1,   2,     1;
  1,   6,     3,     1;
  1,  20,    20,     4,    1;
  1,  70,   175,    50,    5,   1;
  1, 252,  1764,   980,  105,   6, 1;
  1, 924, 19404, 24696, 4116, 196, 7, 1;
  ...

Prog

(PARI) T(n, k) = prod(j=0, k-1, binomial(2*n-2*k+j, n-k)/binomial(n-k+j, j)); \\ Seiichi Manyama, Apr 02 2021

Crossrefs

Row sums give A120259.

Cf. A000891, A000984, A103905, A120257.

Sequence in context: A263341 A201198 A349933 * A201922 A181644 A144351

Adjacent sequences: A120255 A120256 A120257 * A120259 A120260 A120261

Keyword

easy,nonn,tabl

Author

Paul Barry, Jun 13 2006

Status

approved