cp's OEIS Frontend

A086620 Symmetric square table of coefficients, read by antidiagonals, where T(n,k) is the coefficient of x^n*y^k in f(x,y) that satisfies f(x,y) = 1/(1-x-y) + xy*f(x,y)^2.

%I A086620 #7 Mar 30 2012 18:58:53
%S A086620 1,1,1,1,3,1,1,5,5,1,1,7,14,7,1,1,9,28,28,9,1,1,11,47,79,47,11,1,1,13,
%T A086620 71,175,175,71,13,1,1,15,100,331,504,331,100,15,1,1,17,134,562,1196,
%U A086620 1196,562,134,17,1,1,19,173,883,2464,3514,2464,883,173,19,1,1,21,217
%N A086620 Symmetric square table of coefficients, read by antidiagonals, where T(n,k) is the coefficient of x^n*y^k in f(x,y) that satisfies f(x,y) = 1/(1-x-y) + xy*f(x,y)^2.
%C A086620 Determinants of upper left n X n matrices results in A086619: {1,2,10,150,7650,1438200,1051324200,...}, which is the products of the first n terms of the binomial transform of Catalan numbers (A007317): {1,2,5,15,51,188,731,2950,...}.
%F A086620 Contribution from _Paul Barry_, Feb 04 2009: (Start)
%F A086620 T(n,k)=sum{j=0..n+k, C(k,j-k)*C(n+2k-j,k)*if(k<=j,A000108(n-k),0)};
%F A086620 Regarded as a number triangle read by row, columns are generated by sum{j=0..k, C(k,j)*A000108(j)*x^j}*x^k/(1-x)^(k+1). (End)
%e A086620 Rows begin:
%e A086620 1,_1,__1,__1,___1,____1,____1,_____1, ...
%e A086620 1,_3,__5,__7,___9,___11,___13,____15, ...
%e A086620 1,_5,_14,_28,__47,___71,__100,___134, ...
%e A086620 1,_7,_28,_79,_175,__331,__562,___883, ...
%e A086620 1,_9,_47,175,_504,_1196,_2464,__4572, ...
%e A086620 1,11,_71,331,1196,_3514,_8764,_19244, ...
%e A086620 1,13,100,562,2464,_8764,26172,_67740, ...
%e A086620 1,15,134,883,4572,19244,67740,204831, ...
%Y A086620 Cf. A086621 (diagonal), A086622 (antidiagonal sums), A086619 (determinants).
%K A086620 nonn,tabl
%O A086620 0,5
%A A086620 _Paul D. Hanna_, Jul 24 2003