%I #7 Aug 28 2017 07:15:09

%S 1,2,8,56,400,3072,24544,203520,1728256,14967296,131689472,1173936128,

%T 10579907072,96238768128,882437177344,8147574407168,75685465759744,

%U 706854135595008,6633217371029504,62514337980088320,591441701724880896,5615172282703937536,53480608406362914816,510849109679635693568,4892689722718505271296

%N G.f. A(x) satisfies: A( 2*A(x)^2 - 8*A(x)^3 ) = 2*x^2.

%H Paul D. Hanna, <a href="/A291314/b291314.txt">Table of n, a(n) for n = 1..300</a>

%F G.f. A(x) satisfies: A( sqrt( A(2*x^2 - 8*x^3)/2 ) ) = x.

%F a(n) ~ c * d^n / n^(3/2), where d = 10.18955737678412420186395666504... and c = 0.037604336950996597... - _Vaclav Kotesovec_, Aug 28 2017

%e G.f.: A(x) = x + 2*x^2 + 8*x^3 + 56*x^4 + 400*x^5 + 3072*x^6 + 24544*x^7 + 203520*x^8 + 1728256*x^9 + 14967296*x^10 + 131689472*x^11 + 1173936128*x^12 + 10579907072*x^13 + 96238768128*x^14 + 882437177344*x^15 + 8147574407168*x^16 +...

%e such that A( 2*A(x)^2 - 8*A(x)^3 ) = 2*x^2.

%e RELATED SERIES.

%e 2*A(x)^2 - 8*A(x)^3 = 2*x^2 - 8*x^4 - 256*x^8 + 512*x^10 - 10240*x^12 + 69632*x^14 - 835584*x^16 + 7929856*x^18 - 81002496*x^20 + 791674880*x^22 - 8468299776*x^24 + 84863352832*x^26 - 913569742848*x^28 + 9452686147584*x^30 +...

%e Define Ai(x) such that Ai(A(x)) = x, then Ai(x) begins:

%e Ai(x) = x - 2*x^2 - 16*x^4 + 16*x^5 - 160*x^6 + 544*x^7 - 3264*x^8 + 15488*x^9 - 79104*x^10 + 386560*x^11 - 2067456*x^12 + 10359296*x^13 - 55759872*x^14 + 288473088*x^15 - 1546903552*x^16 + 8224194560*x^17 - 44310626304*x^18 + 238776467456*x^19 - 1295524724736*x^20 +...

%e where Ai(x) = sqrt( A(2*x^2 - 8*x^3)/2 )

%e and Ai( 2*Ai(x)^2 ) = 2*x^2 - 8*x^3.

%o (PARI) {a(n) = my(V=[1]); for(i=1, n, V=concat(V, 0); A = x*Ser(V); V[#V] = -polcoeff(subst(G=A, x, 2*A^2 - 8*A^3 ), #V+1)/4); V[n]}

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

%Y Cf. A271961, A291313, A291315.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Aug 22 2017