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A264412 G.f. A(x) satisfies: A(x)^2 = A(x^2) + 6*x.

%I A264412 #18 Apr 16 2016 22:38:41
%S A264412 1,3,-3,9,-33,126,-513,2214,-9876,45045,-209493,990198,-4741191,
%T A264412 22946247,-112079214,551793303,-2735330190,13641353118,-68394016548,
%U A264412 344539469889,-1743035351958,8851923849123,-45110440515753,230615809867476,-1182376529280117,6078184963674498,-31322206517658453,161774639164275552,-837290923919381322
%N A264412 G.f. A(x) satisfies: A(x)^2 = A(x^2) + 6*x.
%H A264412 Paul D. Hanna, <a href="/A264412/b264412.txt">Table of n, a(n) for n = 0..300</a>
%F A264412 Given g.f. A(x), let G(x) denote the g.f. of A264224, then:
%F A264412 (1) G( x/(A(x)^2 - 4*x) ) = x,
%F A264412 (2) G( x/(A(x^2) + 2*x) ) = x,
%F A264412 (3) A(G(x))^2 = (1+4*x) * G(x)/x,
%F A264412 (4) A(G(x)^2) = (1-2*x) * G(x)/x,
%F A264412 where G(x)^2 = G( x^2/(1-4*x) ).
%F A264412 a(n) ~ c * (-1)^(n+1) * d^n / n^(3/2), where d = 5.46806882358680646837..., c = 0.268849330049069376... . - _Vaclav Kotesovec_, Nov 18 2015
%e A264412 G.f.: A(x) = 1 + 3*x - 3*x^2 + 9*x^3 - 33*x^4 + 126*x^5 - 513*x^6 + 2214*x^7 - 9876*x^8 + 45045*x^9 +...
%e A264412 where
%e A264412 A(x)^2 = 1 + 6*x + 3*x^2 - 3*x^4 + 9*x^6 - 33*x^8 + 126*x^10 - 513*x^12 + 2214*x^14 - 9876*x^16 + 45045*x^18 +...
%e A264412 so that A(x)^2 = A(x^2) + 6*x.
%e A264412 Let G(x) = Series_Reversion( x / (A(x^2) + 2*x) ), then
%e A264412 G(x) = x + 2*x^2 + 7*x^3 + 26*x^4 + 103*x^5 + 422*x^6 + 1774*x^7 + 7604*x^8 + 33109*x^9 + 146042*x^10 +...+ A264224(n)*x^n +...
%e A264412 such that G(x)^2 = G( x^2/(1-4*x) ) and A(G(x))^2 = (1+4*x) * G(x)/x.
%o A264412 (PARI) {a(n) = my(A=1); for(i=1,n, A = sqrt( subst(A,x,x^2) + 6*x +x*O(x^n))); polcoeff(A,n)}
%o A264412 for(n=0,30,print1(a(n),", "))
%Y A264412 Cf. A271930, A264224, A264413, A264414, A264415.
%K A264412 sign
%O A264412 0,2
%A A264412 _Paul D. Hanna_, Nov 12 2015