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%I #16 Sep 30 2019 07:15:42
%S 1,3,30,447,8202,171846,3956796,97916895,2567551890,70655670690,
%T 2026596875268,60282027684678,1852444347792036,58633762133405100,
%U 1907098496516434680,63620675921801106495,2173457638433471757282,75940916632597398212298,2710857429948875567968692,98775527832178103444182722,3670845430153146908693608044,139047871842184594320103381524,5365224711989826990651317756232
%N G.f. satisfies: A(x - 2*A(x)^2) = x + A(x)^2.
%H Paul D. Hanna, <a href="/A276361/b276361.txt">Table of n, a(n) for n = 1..300</a>
%F G.f. A(x) also satisfies:
%F (1) A(x) = x + 3 * A( x/3 + 2*A(x)/3 )^2.
%F (2) 2*A(x) = -x + 3 * Series_Reversion(x - 2*A(x)^2).
%F (3) R(x) = -2*x + 3 * Series_Reversion(x + A(x)^2), where R(A(x)) = x.
%F (4) R( sqrt( x/3 - R(x)/3 ) ) = 2*x/3 + R(x)/3, where R(A(x)) = x.
%F a(n) = Sum_{k=0..n-1} A277295(n,k)*2^k*3^(n-k-1).
%e G.f.: A(x) = x + 3*x^2 + 30*x^3 + 447*x^4 + 8202*x^5 + 171846*x^6 + 3956796*x^7 + 97916895*x^8 + 2567551890*x^9 + 70655670690*x^10 + 2026596875268*x^11 + 60282027684678*x^12 +...
%e such that A(x - 2*A(x)^2) = x + A(x)^2.
%e RELATED SERIES.
%e Note that Series_Reversion(x - 2*A(x)^2) = x/3 + 2*A(x)/3, which begins:
%e Series_Reversion(x - 2*A(x)^2) = x + 2*x^2 + 20*x^3 + 298*x^4 + 5468*x^5 + 114564*x^6 + 2637864*x^7 + 65277930*x^8 + 1711701260*x^9 + 47103780460*x^10 + 1351064583512*x^11 + 40188018456452*x^12 +...
%e Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x,
%e R(x) = x - 3*x^2 - 12*x^3 - 132*x^4 - 1992*x^5 - 36144*x^6 - 742176*x^7 - 16688880*x^8 - 402824928*x^9 - 10300868160*x^10 - 276531035520*x^11 - 7742210941056*x^12 +...
%e then Series_Reversion(x + A(x)^2) = 2*x/3 + R(x)/3.
%t m = 24; A[_] = 0;
%t Do[A[x_] = x + 3 A[x/3 + 2 A[x]/3]^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-2*F^2) - F^2, #A) ); A[n]}
%o for(n=1, 30, print1(a(n), ", "))
%Y Cf. A277295, A213591, A275765, A276360, A276362, A276363.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Aug 31 2016
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