A131429 Triangle read by rows: T(n,k) = C(n) + C(k) - 1 where C(n) = A000108(n) are the Catalan numbers, 0 <= k <= n.

1, 1, 1, 2, 2, 3, 5, 5, 6, 9, 14, 14, 15, 18, 27, 42, 42, 43, 46, 55, 83, 132, 132, 133, 136, 145, 173, 263, 429, 429, 430, 433, 442, 470, 560, 857, 1430, 1430, 1431, 1434, 1443, 1471, 1561, 1858, 2859, 4862, 4862, 4863, 4866, 4875, 4903, 4993, 5290, 6291, 9723

Offset: 0

Author

Gary W. Adamson, Jul 10 2007

Keywords

• nonn
• tabl

Comments

Left column = Catalan numbers, A000108. Right border = 2*A000108 - 1. Row sums = A131430: (1, 2, 7, 25, 88, 311, 1114, ...).

Examples

			First few rows of the triangle are:
   1;
   1,  1;
   2,  2,  3;
   5,  5,  6,  9;
  14, 14, 15, 18, 27;
  42, 42, 43, 46, 55, 83;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1274 (first 50 rows)

Crossrefs

Column k=0 is A000108.

Row sums are A131430.

Cf. A131427, A131428.

Programs

PARI

T(n,k)=if(k<=n, binomial(2*n,n)/(n+1) + binomial(2*k,k)/(k+1) - 1, 0) \\ Andrew Howroyd, Sep 01 2018

Formula

Equals (A000012 * A131427) + (A131427 * A000012) - A000012 as infinite lower triangular matrices.

Extensions

Name clarified by Andrew Howroyd, Sep 01 2018