%I #8 Nov 23 2017 06:21:13
%S 1,1,1,5,11,41,110,393,1180,4199,13538,48353,163130,585790,2035789,
%T 7353701,26078670,94767692,340963400,1246208999,4531981344,
%U 16653522634,61062317840,225484886305,832165390554,3086496104906,11451263408809,42640078377086,158896249065150,593747065988969,2220821612331954,8324534184912813,31236600473715868,117415053041080870
%N G.f. A(x) satisfies: A(x) = 1 + x*A(x)^2 - x^2/A(x)^2.
%C Note that G(x) such that G(x) = 1 + x*G(x)^2 - x^2/G(x)^4 has negative coefficients.
%H Paul D. Hanna, <a href="/A295504/b295504.txt">Table of n, a(n) for n = 0..500</a>
%F G.f. A(x) satisfies: x^2 = A(x)^2 - A(x)^3 + x*A(x)^4.
%F a(n) ~ s * sqrt((2*r - s^4)/(Pi*(3*r - s^4))) / (2*n^(3/2)*r^n), where r = 0.2543332033061789613545588837164265808807851525806... and s = 1.93057898402081849972124794072346460422213155223... are real roots of the system of equations 1 + r*s^2 = r^2/s^2 + s, 2*r*(r + s^4)/s^3 = 1. - _Vaclav Kotesovec_, Nov 23 2017
%e G.f.: A(x) = 1 + x + x^2 + 5*x^3 + 11*x^4 + 41*x^5 + 110*x^6 + 393*x^7 + 1180*x^8 + 4199*x^9 + 13538*x^10 + 48353*x^11 + 163130*x^12 + 585790*x^13 + 2035789*x^14 + 7353701*x^15 + 26078670*x^16 + 94767692*x^17 + 340963400*x^18 + 1246208999*x^19 + 4531981344*x^20 +...
%e such that A(x) = 1 + x*A(x)^2 - x^2/A(x)^2.
%e RELATED SERIES.
%e 1/A(x)^2 = 1 - 2*x + x^2 - 8*x^3 + 4*x^4 - 44*x^5 + 18*x^6 - 302*x^7 + 8*x^8 - 2488*x^9 - 1484*x^10 - 23472*x^11 - 30265*x^12 +...
%e A(x)^2 = 1 + 2*x + 3*x^2 + 12*x^3 + 33*x^4 + 114*x^5 + 349*x^6 + 1198*x^7 + 3897*x^8 + 13546*x^9 + 45865*x^10 + 161646*x^11 + 562318*x^12 +...
%e A(x)^3 = 1 + 3*x + 6*x^2 + 22*x^3 + 69*x^4 + 237*x^5 + 781*x^6 + 2694*x^7 + 9165*x^8 + 32051*x^9 + 111633*x^10 + 395724*x^11 + 1402514*x^12 +...
%e A(x)^4 = 1 + 4*x + 10*x^2 + 36*x^3 + 123*x^4 + 432*x^5 + 1496*x^6 + 5268*x^7 + 18505*x^8 + 65768*x^9 + 234078*x^10 + 840196*x^11 + 3023661*x^12 +...
%e where x^2 = A(x)^2 - A(x)^3 + x*A(x)^4.
%e Let F(x) be the series given by
%e F(x) = (1/x)*Series_Reversion(x*A(x)) = 1 - x + x^2 - 5*x^3 + 15*x^4 - 55*x^5 + 220*x^6 - 876*x^7 + 3645*x^8 - 15485*x^9 + 66735*x^10 +...+ (-1)^n*A137959(n)*x^n +...
%e then F(x) = 1 - x + x^2*F(x)^5.
%o (PARI) {a(n) = local(A=1+x); for(i=1, n, A = 1 + x*A^2 - x^2/A^2 +x*O(x^n)); polcoeff(G=A, n)}
%o for(n=0,40,print1(a(n),", "))
%Y Cf. A137959, A295404, A295498.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Nov 22 2017