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A089449 Antidiagonal sums of square table A089447, which lists the coefficients 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 A089449 #8 Oct 10 2020 04:51:21
%S A089449 1,2,6,22,90,396,1837,8870,44186,225628,1175322,6222788,33392644,
%T A089449 181216728,992829379,5483790870,30502513970,170705626308,960498281302,
%U A089449 5430200987260,30830681187480,175715526842056,1004931956037782
%N A089449 Antidiagonal sums of square table A089447, which lists the coefficients 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.
%H A089449 Vaclav Kotesovec, <a href="/A089449/b089449.txt">Table of n, a(n) for n = 0..1000</a>
%F A089449 G.f.: A(x) = sum(n>=0, Catalan(n+1)*x^n) + x^2*A(x)^4, where Catalan(n)=(2n)!/(n!*(n+1)!).
%F A089449 From _Vaclav Kotesovec_, Oct 10 2020: (Start)
%F A089449 G.f.: A(x) = (1 - Sqrt[1-4*x] - 2*x)/(2*x^2) + x^2*A(x)^4.
%F A089449 a(n) ~ sqrt(11) * 3^(15/2 + 3*n) / ((8 + 3*sqrt(3) + 4*sqrt(4 + 3*sqrt(3))) * sqrt((2519 + 528*sqrt(3) + 2*sqrt(1484692 + 881529*sqrt(3))) * Pi) * n^(3/2) * 2^(2*n + 5/2) * (-32 - 18*sqrt(3) + sqrt(1996 + 1233*sqrt(3)))^(n+2)). (End)
%Y A089449 Cf. A089447 (table), A089448 (diagonal), A002293.
%K A089449 nonn
%O A089449 0,2
%A A089449 _Paul D. Hanna_, Nov 02 2003