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A115253 "Correlation triangle" for Catalan numbers.

%I A115253 #8 Jul 11 2020 17:18:26
%S A115253 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,
%T A115253 132,429,174,103,83,83,103,174,429,1430,561,320,245,227,245,320,561,
%U A115253 1430,4862,1859,1032,763,671,671,763,1032,1859,4862,16796,6292,3421
%N A115253 "Correlation triangle" for Catalan numbers.
%C A115253 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).
%F A115253 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).
%F A115253 T(n, k) = sum{j=0..n, [j<=k]*C(k-j)[j<=n-k]*C(n-k-j)}.
%F A115253 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
%e A115253 Triangle begins
%e A115253 1;
%e A115253 1, 1;
%e A115253 2, 2, 2;
%e A115253 5, 3, 3, 5;
%e A115253 14, 7, 6, 7, 14;
%e A115253 42, 19, 13, 13, 19, 42;
%K A115253 easy,nonn,tabl
%O A115253 0,4
%A A115253 _Paul Barry_, Jan 18 2006