A115253 "Correlation triangle" for Catalan numbers.

1, 1, 1, 2, 2, 2, 5, 3, 3, 5, 14, 7, 6, 7, 14, 42, 19, 13, 13, 19, 42, 132, 56, 35, 31, 35, 56, 132, 429, 174, 103, 83, 83, 103, 174, 429, 1430, 561, 320, 245, 227, 245, 320, 561, 1430, 4862, 1859, 1032, 763, 671, 671, 763, 1032, 1859, 4862, 16796, 6292, 3421

Offset: 0

Author

Paul Barry, Jan 18 2006

Keywords

• easy
• nonn
• tabl

Comments

Row sums are A094639. Diagonal sums are A115254. Corresponds to the triangle of antidiagonals of the correlation matrix of the sequence array for C(n).

Examples

			Triangle begins
1;
1, 1;
2, 2, 2;
5, 3, 3, 5;
14, 7, 6, 7, 14;
42, 19, 13, 13, 19, 42;
		

Formula

G.f.: c(x)c(x*y)/(1-x^2*y) where c(x) is the g.f. of A000108 (format due to Christian G. Bower).

T(n, k) = sum{j=0..n, [j<=k]*C(k-j)[j<=n-k]*C(n-k-j)}.

O.g.f.: F(z,v) = 1/4 ((1-sqrt(1-4 z)) (1-sqrt(-4 v z+1)))/(z^2 v (-v z^2+1)). - Yu-Sheng Chang, Jun 12 2020