A089449 - OEIS

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 #8 Oct 10 2020 04:51:21

%S 1,2,6,22,90,396,1837,8870,44186,225628,1175322,6222788,33392644,

%T 181216728,992829379,5483790870,30502513970,170705626308,960498281302,

%U 5430200987260,30830681187480,175715526842056,1004931956037782

%N 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 Vaclav Kotesovec, <a href="/A089449/b089449.txt">Table of n, a(n) for n = 0..1000</a>

%F 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 From _Vaclav Kotesovec_, Oct 10 2020: (Start)

%F G.f.: A(x) = (1 - Sqrt[1-4*x] - 2*x)/(2*x^2) + x^2*A(x)^4.

%F 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 Cf. A089447 (table), A089448 (diagonal), A002293.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Nov 02 2003