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%I #3 Mar 30 2012 18:36:38
%S 1,1,1,1,1,1,1,2,2,1,1,3,4,3,1,1,4,8,8,4,1,1,5,14,19,14,5,1,1,6,22,40,
%T 40,22,6,1,1,7,32,76,100,76,32,7,1,1,8,44,132,222,222,132,44,8,1,1,9,
%U 58,213,448,570,448,213,58,9,1,1,10,74,324,834,1316,1316,834,324,74,10,1,1
%N 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-xy)/[(1-x)(1-y)] + xy*f(x,y)^2.
%C The first row and column of 1's together form: (1-xy)/[(1-x)(1-y)], while the remaining square table (excluding the first row and column) give the coefficients of f(x,y)^2.
%e Rows begin:
%e 1,1,_1,__1,___1,___1,____1,____1,_____1, ...
%e 1,1,_2,__3,___4,___5,____6,____7,_____8, ...
%e 1,2,_4,__8,__14,__22,___32,___44,____58, ...
%e 1,3,_8,_19,__40,__76,__132,__213,___324, ...
%e 1,4,14,_40,_100,_222,__448,__834,__1450, ...
%e 1,5,22,_76,_222,_570,_1316,_2782,__5458, ...
%e 1,6,32,132,_448,1316,_3442,_8180,_17928, ...
%e 1,7,44,213,-834,2782,_8180,21685,_52694, ...
%e 1,8,58,324,1450,5458,17928,52694,141112, ...
%Y Cf. A086624 (diagonal), A086625 (antidiagonal sums).
%K nonn,tabl
%O 0,8
%A _Paul D. Hanna_, Jul 24 2003
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