%I #18 Sep 30 2019 07:15:50
%S 1,4,40,564,9592,184008,3844624,85700980,2011283640,49248127800,
%T 1250064156912,32736194249256,881252194701616,24317581366876880,
%U 686300288661644960,19774058901706750100,580795172081872246232,17368587281321383296184,528294942152813411073968,16329939570298980826852824,512590568042639978453793744,16329084800479729420462546352,527621994750854274463428080608
%N G.f. satisfies: A(x - A(x)^2) = x + 3*A(x)^2.
%H Paul D. Hanna, <a href="/A276362/b276362.txt">Table of n, a(n) for n = 1..300</a>
%F G.f. A(x) also satisfies:
%F (1) A(x) = x + 4 * A( 3*x/4 + A(x)/4 )^2.
%F (2) A(x) = -3*x + 4 * Series_Reversion(x - A(x)^2).
%F (3) 3*R(x) = -x + 4 * Series_Reversion(x + 3*A(x)^2), where R(A(x)) = x.
%F (4) R( sqrt( x/3 - R(x)/3 ) ) = x/4 + 3*R(x)/4, where R(A(x)) = x.
%F a(n) = Sum_{k=0..n-1} A277295(n,k)*4^(n-k-1).
%e G.f.: A(x) = x + 4*x^2 + 40*x^3 + 564*x^4 + 9592*x^5 + 184008*x^6 + 3844624*x^7 + 85700980*x^8 + 2011283640*x^9 + 49248127800*x^10 + 1250064156912*x^11 + 32736194249256*x^12 +...
%e such that A(x - A(x)^2) = x + 3*A(x)^2.
%e RELATED SERIES.
%e Note that Series_Reversion(x - A(x)^2) = 3*x/4 + A(x)/4, which begins:
%e Series_Reversion(x - A(x)^2) = x + x^2 + 10*x^3 + 141*x^4 + 2398*x^5 + 46002*x^6 + 961156*x^7 + 21425245*x^8 + 502820910*x^9 + 12312031950*x^10 + 312516039228*x^11 + 8184048562314*x^12 +...
%e Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x,
%e R(x) = x - 4*x^2 - 8*x^3 - 84*x^4 - 1112*x^5 - 17352*x^6 - 303824*x^7 - 5791060*x^8 - 117898648*x^9 - 2531645240*x^10 - 56835852080*x^11 - 1325547044072*x^12 +...
%e then Series_Reversion(x + 3*A(x)^2) = x/4 + 3*R(x)/4.
%t m = 24; A[_] = 0;
%t Do[A[x_] = x + 4 A[3x/4 + A[x]/4]^2 + O[x]^m // Normal, {m}];
%t CoefficientList[A[x]/x, x] (* _Jean-François Alcover_, Sep 30 2019 *)
%o (PARI) {a(n) = my(A=[1],F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x - F^2) - 3*F^2, #A) ); A[n]}
%o for(n=1, 30, print1(a(n), ", "))
%Y Cf. A277295, A213591, A275765, A276360, A276361, A276363.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Aug 31 2016