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Slanted Catalan convolution table, read by rows of 2*n+1 terms in row n, where T(n,k) = C(n+2*k-[k/2],k)*(n-[k/2])/(n+2*k-[k/2]).
2

%I #13 Feb 12 2023 12:34:14

%S 1,1,1,0,1,2,2,5,0,1,3,5,14,14,42,0,1,4,9,28,42,132,132,429,0,1,5,14,

%T 48,90,297,429,1430,1430,4862,0,1,6,20,75,165,572,1001,3432,4862,

%U 16796,16796,58786,0,1,7,27,110,275,1001,2002,7072,11934,41990,58786,208012

%N Slanted Catalan convolution table, read by rows of 2*n+1 terms in row n, where T(n,k) = C(n+2*k-[k/2],k)*(n-[k/2])/(n+2*k-[k/2]).

%C Row sums form A100248. Antidiagonal sums form A100249.

%F T(n, k) = A033184(n-[k/2], k) for n>0 (with A033184 formatted as a square array).

%F G.f. A(x, y) satisfies:

%F A(x^2, y) = ( (1+x)/(2*y - x*(1 - sqrt(1 - 4*x*y))) - (1-x)/(2*y + x*(1 - sqrt(1 + 4*x*y))) )*y/x.

%e Rows begin:

%e [1],

%e [1,1,0],

%e [1,2,2,5,0],

%e [1,3,5,14,14,42,0],

%e [1,4,9,28,42,132,132,429,0],

%e [1,5,14,48,90,297,429,1430,1430,4862,0],

%e [1,6,20,75,165,572,1001,3432,4862,16796,16796,58786,0],...

%e and is derived from the square array of Catalan convolutions (A033184)

%e by shifting each column k down by [k/2] rows.

%o (PARI) {T(n,k) = if(n==k&k==0,1,binomial(n+2*k-(k\2),k)*(n-(k\2))/(n+2*k-(k\2)))}

%o for(n=0,10,for(k=0,2*n, print1(T(n,k),", "));print(""))

%o (PARI) {T(n,k) = polcoeff(((1-sqrt(1-4*z +z*O(z^(2*n))))/(2*z))^(n-k\2),k,z)}

%o for(n=0,10,for(k=0,2*n, print1(T(n,k),", "));print(""))

%Y Cf. A033184, A100248, A100249.

%K nonn,tabf

%O 0,6

%A _Paul D. Hanna_, Nov 09 2004