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

0,4

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).

Links

Table of n, a(n) for n=0..57.

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

Example

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

Crossrefs

Sequence in context: A061311 A174960 A210239 * A154429 A333388 A174577

Adjacent sequences: A115250 A115251 A115252 * A115254 A115255 A115256

Keyword

easy,nonn,tabl

Author

Paul Barry, Jan 18 2006

Status

approved