A128562 Triangle, read by rows, where T(n,k) is the coefficient of q^((n+1)*k) in the q-binomial coefficient [2*n+1, n] for n >= k >= 0.

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 6, 12, 6, 1, 1, 10, 29, 29, 10, 1, 1, 14, 61, 94, 61, 14, 1, 1, 21, 120, 263, 263, 120, 21, 1, 1, 29, 222, 645, 910, 645, 222, 29, 1, 1, 41, 392, 1468, 2724, 2724, 1468, 392, 41, 1, 1, 55, 669, 3113, 7352, 9686, 7352, 3113, 669, 55, 1

Offset: 0

Author

Paul D. Hanna, Mar 10 2007

Keywords

• nonn
• tabl

Comments

Row sums equal a shifted version of A003239 (number of rooted planar trees with n non-root nodes). Column 1 is a shifted version of A000065 (-1 + number of partitions of n). Column 2 is a shifted version of A128563. This array is a variant of triangles A128545 and A047812 (Parker's partition triangle).

Examples

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  1,  1;
  1,  2,   1;
  1,  4,   4,    1;
  1,  6,  12,    6,    1;
  1, 10,  29,   29,   10,    1;
  1, 14,  61,   94,   61,   14,    1;
  1, 21, 120,  263,  263,  120,   21,    1;
  1, 29, 222,  645,  910,  645,  222,   29,   1;
  1, 41, 392, 1468, 2724, 2724, 1468,  392,  41,  1;
  1, 55, 669, 3113, 7352, 9686, 7352, 3113, 669, 55, 1;
  ...
		

Crossrefs

Cf. A000065 (column 1), A003239 (row sums), A128563 (column 2).

Variants are A047812 and A128545.

Programs

PARI

T(n,k)=if(n
				

Formula

T(n,k) = [q^((n+1)*k)] Product_{j=n+1..2*n+1}(1-q^j) / Product_{j=1..n+1}(1-q^j).

Extensions

Minor edits by Petros Hadjicostas, Jun 01 2020