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%I #5 Jul 19 2016 22:38:17
%S 1,3,24,225,2451,28584,350811,4456971,58132194,773773785,10468458657,
%T 143528736888,1989864432072,27848242441521,392899157668962,
%U 5582238744258009,79799757710924847,1146961633262521734,16564959319032187542,240274065224349972819,3498743054027725572015,51126484223473738706979,749497753666229701655097,11019579243648283996016040,162451851753106862734656771,2400803315482934131290808344
%N G.f. A(x) satisfies: A(x)^2 - 6*A(x)^3 = A(x^2).
%H Paul D. Hanna, <a href="/A274737/b274737.txt">Table of n, a(n) for n = 1..300</a>
%F G.f. A(x) satisfies: A(B(x)^2) = x^2 - 6*x^3, where A(B(x)) = x.
%e G.f.: A(x) = x + 3*x^2 + 24*x^3 + 225*x^4 + 2451*x^5 + 28584*x^6 + 350811*x^7 + 4456971*x^8 + 58132194*x^9 + 773773785*x^10 +...
%e such that A(x)^2 - 6*A(x)^3 = A(x^2).
%e RELATED SERIES.
%e A(x)^2 = x^2 + 6*x^3 + 57*x^4 + 594*x^5 + 6828*x^6 + 82674*x^7 + 1041399*x^8 + 13493790*x^9 + 178715343*x^10 + 2408259060*x^11 + 32912262864*x^12 +...
%e A(x)^3 = x^3 + 9*x^4 + 99*x^5 + 1134*x^6 + 13779*x^7 + 173529*x^8 + 2248965*x^9 + 29785482*x^10 + 401376510*x^11 + 5485372380*x^12 +...
%e Let B(x) denote the series reversion of g.f. A(x), so that A(B(x)) = x, where
%e B(x) = x - 3*x^2 - 6*x^3 - 75*x^5 - 171*x^6 - 1287*x^7 - 4239*x^8 - 23289*x^9 - 107001*x^10 - 585468*x^11 - 2852334*x^12 - 15659352*x^13 - 80867160*x^14 +...
%e Note that g.f. A(x) and B(x) satisfy:
%e (1) A(B(x)^2) = C(x) = x^2 - 6*x^3.
%e (2) A(B(x)^4) = C(C(x)) = x^4 - 12*x^5 + 30*x^6 + 108*x^7 - 648*x^8 + 1296*x^9.
%e (3) A(B(x)^8) = C(C(C(x))).
%o (PARI) /* From A(B(x)^2) = x^2 - 6*x^3, where A(B(x)) = x: */
%o {a(n) = my(A=[1, 3], F, B); for(i=1, n, A=concat(A, 0); F=x*Ser(A); B=serreverse(F); A[#A] = Vec(subst(F, x, B^2))[#A]/2); A[n]}
%o for(n=1, 30, print1(a(n), ", "))
%Y Cf. A273095, A274736.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jul 19 2016
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