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A089447 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) = g(x,y) + xy*f(x,y)^4 and where g(x,y) satisfies: 1 + (x+y-1)*g(x,y) + xy*g(x,y)^2 = 0.

%I A089447 #15 Jun 07 2016 10:27:57
%S A089447 1,1,1,1,4,1,1,10,10,1,1,20,48,20,1,1,35,162,162,35,1,1,56,441,841,
%T A089447 441,56,1,1,84,1036,3314,3314,1036,84,1,1,120,2184,10786,18004,10786,
%U A089447 2184,120,1,1,165,4236,30460,77952,77952,30460,4236,165,1,1,220,7689,77044
%N A089447 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) = g(x,y) + xy*f(x,y)^4 and where g(x,y) satisfies: 1 + (x+y-1)*g(x,y) + xy*g(x,y)^2 = 0.
%C A089447 Explicitly, g(x,y) = ((1-x-y)+sqrt((1-x-y)^2-4xy))/(2xy) = sum(n>=0, sum(k>=0, N(n,k)*x^n*y^k), where N(n,k) are the Narayana numbers: N(n,k) = C(n+k,k)*C(n+k+2,k+1)/(n+k+2). This array is directly related to sequence A002293, which has a g.f. h(x) that satisfies h(x) = 1 + x*h(x)^4. The inverse binomial transform of the rows grows by three terms per row.
%e A089447 Rows begin:
%e A089447 [1,   1,     1,      1,       1,        1,         1,          1, ...];
%e A089447 [1,   4,    10,     20,      35,       56,        84,        120, ...];
%e A089447 [1,  10,    48,    162,     441,     1036,      2184,       4236, ...];
%e A089447 [1,  20,   162,    841,    3314,    10786,     30460,      77044, ...];
%e A089447 [1,  35,   441,   3314,   18004,    77952,    284880,     912042, ...];
%e A089447 [1,  56,  1036,  10786,   77952,   435654,   2007456,    7951674, ...];
%e A089447 [1,  84,  2184,  30460,  284880,  2007456,  11427992,   55009548, ...];
%e A089447 [1, 120,  4236,  77044,  912042,  7951674,  55009548,  317112363, ...];
%e A089447 [1, 165,  7689, 178387, 2624453, 27870393, 231114465, 1576219474, ...]; ...
%o A089447 (PARI) {L=10; T=matrix(L,L,n,k,1); for(n=1,L-1, for(k=1,L-1, T[n+1,k+1]=binomial(n+k,k)*binomial(n+k+2,k+1)/(n+k+2)+ sum(j3=1,k,sum(i3=1,n,T[n-i3+1,k-j3+1]* sum(j2=1,j3,sum(i2=1,i3,T[i3-i2+1,j3-j2+1]* sum(j1=1,j2,sum(i1=1,i2,T[i2-i1+1,j2-j1+1]*T[i1,j1])); )); )); )); T}
%Y A089447 Cf. A089448 (diagonal), A089449 (antidiagonal sums), A086617, A088925, A002293.
%K A089447 nonn,tabl
%O A089447 0,5
%A A089447 _Paul D. Hanna_, Nov 02 2003