%I #11 Aug 28 2017 07:39:11
%S 1,-2,6,-26,100,-460,2258,-11558,60786,-326826,1785930,-9893778,
%T 55447800,-313817720,1791442406,-10303155322,59642852324,
%U -347233450156,2031756438046,-11941773701426,70471288256196,-417379686511812,2480161711278070,-14781955283569090,88343937381017274,-529319474378769346,3178848917169132254,-19131855254581689246
%N G.f. A(x) satisfies: A( 2*A(x)^2 + 4*A(x)^3 ) = 2*x^2 - 4*x^3.
%C Series reversion of the g.f. is described by A290957.
%H Paul D. Hanna, <a href="/A290958/b290958.txt">Table of n, a(n) for n = 1..520</a>
%F a(n) ~ (-1)^(n+1) * c * d^n / n^(3/2), where d = 6.36304571910819028529344... and c = 0.086619593102483539978... - _Vaclav Kotesovec_, Aug 28 2017
%e G.f.: A(x) = x - 2*x^2 + 6*x^3 - 26*x^4 + 100*x^5 - 460*x^6 + 2258*x^7 - 11558*x^8 + 60786*x^9 - 326826*x^10 + 1785930*x^11 - 9893778*x^12 + 55447800*x^13 - 313817720*x^14 + 1791442406*x^15 - 10303155322*x^16 + 59642852324*x^17 - 347233450156*x^18 + 2031756438046*x^19 - 11941773701426*x^20 +...
%e such that A( 2*A(x)^2 - 4*A(x)^3 ) = 2*x^2 + 4*x^3.
%e Let B(x) be the series reversion of A(x), then B(x) is the g.f. of A290957 and begins;
%e B(x) = x + 2*x^2 + 2*x^3 + 6*x^4 + 40*x^5 + 208*x^6 + 798*x^7 + 3122*x^8 + 15038*x^9 + 77830*x^10 + 381798*x^11 + 1819998*x^12 + 8925172*x^13 + 45280900*x^14 + 231030138*x^15 + 1171823534*x^16 +...+ A290957(n)*x^n +...
%e where B( 2*x^2 - 4*x^3 ) = 2*A(x)^2 + 4*A(x)^3,
%e also, A( 2*x^2 + 4*x^3 ) = 2*B(x)^2 - 4*B(x)^3,
%e and B( 2*B(x)^2 - 4*B(x)^3 ) = 2*x^2 + 4*x^3.
%e Related series begin:
%e 2*A(x)^2 + 4*A(x)^3 = 2*x^2 + 12*x^3 + 40*x^4 + 112*x^5 + 416*x^6 + 2112*x^7 + 10336*x^8 + 45936*x^9 + 206192*x^10 + 999376*x^11 + 5026640*x^12 +...
%e 2*B(x)^2 - 4*B(x)^3 = 2*x^2 - 12*x^3 + 56*x^4 - 272*x^5 + 1312*x^6 - 6432*x^7 + 32640*x^8 - 170576*x^9 + 911696*x^10 - 4963760*x^11 + 27425200*x^12 +...
%o (PARI) /* Informal code to generate N terms */
%o {C=[1,-2]; for(i=1,N=60,
%o A = sum(n=1,#C,C[n]*x^(n) ) + t*x^(#C+1) +O(x^(#C+2));
%o S = subst(A,x, 2*A^2 + 4*A^3);
%o C = concat(C,polcoeff(subst(-S/deriv(polcoeff(S,#C+2,x),t),t,0),#C+2,x) )); C}
%Y Cf. A290957 (inverse), A271961.
%K sign
%O 1,2
%A _Paul D. Hanna_, Aug 14 2017