%I #11 Apr 17 2016 06:44:42

%S 1,10,-45,450,-5535,75600,-1106100,16953750,-268652880,4365638550,

%T -72354858300,1218356280000,-20784495119850,358457180010750,

%U -6239532583193625,109476057598087500,-1934128026918961515,34378012275668994150,-614328464414815220025,11030366153872043358750,-198899407327466712808800,3600377821710426377668500

%N G.f. A(x) satisfies: A(x)^2 = A(x^2) + 20*x.

%H Paul D. Hanna, <a href="/A264414/b264414.txt">Table of n, a(n) for n = 0..300</a>

%F Given g.f. A(x), let G(x) denote the g.f. of A264226, then:

%F (1) G( x/(A(x)^2 - 16*x) ) = x,

%F (2) G( x/(A(x^2) + 4*x) ) = x,

%F (3) A(G(x))^2 = (1+16*x) * G(x)/x,

%F (4) A(G(x)^2) = (1-4*x) * G(x)/x,

%F where G(x)^2 = G( x^2/(1-8*x) ).

%e G.f.: A(x) = 1 + 10*x - 45*x^2 + 450*x^3 - 5535*x^4 + 75600*x^5 - 1106100*x^6 +...

%e where

%e A(x)^2 = 1 + 20*x + 10*x^2 - 45*x^4 + 450*x^6 - 5535*x^8 + 75600*x^10 - 1106100*x^12 +...

%e so that A(x)^2 = A(x^2) + 20*x.

%e Let G(x) = Series_Reversion( x / (A(x^2) + 4*x) ), then

%e G(x) = x + 4*x^2 + 28*x^3 + 208*x^4 + 1702*x^5 + 14584*x^6 + 129808*x^7 + 1187008*x^8 + 11089153*x^9 + 105370660*x^10 +...+ A264226(n)*x^n +...

%e such that G(x)^2 = G( x^2/(1-8*x) ).

%o (PARI) {a(n) = my(A=1); for(i=1,n, A = sqrt( subst(A,x,x^2) + 20*x +x*O(x^n))); polcoeff(A,n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A271957, A264226, A264412, A264413, A264415.

%K sign

%O 0,2

%A _Paul D. Hanna_, Nov 12 2015