%I A100247
%S A100247 1,1,1,0,1,2,2,5,0,1,3,5,14,14,42,0,1,4,9,28,42,132,132,429,0,1,5,14,48,
%T A100247 90,297,429,1430,1430,4862,0,1,6,20,75,165,572,1001,3432,4862,16796,
%U A100247 16796,58786,0,1,7,27,110,275,1001,2002,7072,11934,41990,58786,208012
%N A100247 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 A100247 Row sums form A100248. Antidiagonal sums form A100249.
%F A100247 T(n, k) = A033184(n-[k/2], k) for n>0 (with A033184 formatted as a square 
               array). G.f. A(x, y) satisfies: 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 A100247 Rows begin:
%e A100247 [1],
%e A100247 [1,1,0],
%e A100247 [1,2,2,5,0],
%e A100247 [1,3,5,14,14,42,0],
%e A100247 [1,4,9,28,42,132,132,429,0],
%e A100247 [1,5,14,48,90,297,429,1430,1430,4862,0],
%e A100247 [1,6,20,75,165,572,1001,3432,4862,16796,16796,58786,0],...
%e A100247 and is derived from the square array of Catalan convolutions (A033184)
%e A100247 by shifting each column k down by [k/2] rows.
%o A100247 (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)))} 
               (PARI) {T(n,k)=polcoeff(((1-sqrt(1-4*z))/(2*z))^(n-k\2),k,z)}
%Y A100247 Cf. A033184, A100248, A100249.
%Y A100247 Sequence in context: A037010 A114695 A134084 this_sequence A011342 A084046 
               A016586
%Y A100247 Adjacent sequences: A100244 A100245 A100246 this_sequence A100248 A100249 
               A100250
%K A100247 nonn,tabl
%O A100247 0,6
%A A100247 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 09 2004