%I #12 Sep 30 2019 07:16:18

%S 1,5,60,1000,19970,448160,10926360,283651245,7740058300,220046970860,

%T 6476695275680,196438030797880,6117627849485360,195082685133612800,

%U 6355848358118392400,211189970909192038500,7146354688384980282000,245970478274041025623200,8602606263466490521359400,305460999044315834902424200,11003870605124169641012461600

%N G.f. satisfies: A(x - A(x)^2) = x + 4*A(x)^2.

%H Paul D. Hanna, <a href="/A277300/b277300.txt">Table of n, a(n) for n = 1..300</a>

%F G.f. A(x) also satisfies:

%F (1) A(x) = x + 5 * A( 4*x/5 + A(x)/5 )^2.

%F (2) A(x) = -4*x + 5 * Series_Reversion(x - A(x)^2).

%F (3) R(x) = -x/4 + 5/4 * Series_Reversion(x + 4*A(x)^2), where R(A(x)) = x.

%F (4) R( sqrt( x/5 - R(x)/5 ) ) = x/5 + 4*R(x)/5, where R(A(x)) = x.

%F a(n) = Sum_{k=0..n-1} A277295(n,k) * 5^(n-k-1).

%e G.f.: A(x) = x + 5*x^2 + 60*x^3 + 1000*x^4 + 19970*x^5 + 448160*x^6 + 10926360*x^7 + 283651245*x^8 + 7740058300*x^9 + 220046970860*x^10 +...

%t m = 22; A[_] = 0;

%t Do[A[x_] = x + 5 A[4x/5 + A[x]/5]^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) - 4*F^2, #A) ); A[n]}

%o for(n=1, 30, print1(a(n), ", "))

%Y Cf. A277295, A213591, A275765, A276360, A276361, A276362, A276363.

%Y Cf. A277301, A277302, A277303, A277304, A277305, A277306, A277307, A277308, A277309.

%Y Cf. A276364.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Oct 09 2016