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 A206405 E.g.f. A(x) satisfies: exp(A(x)) = x + 2*exp(A(x)^2) - exp(A(x)^3), with A(0) = 0. 4

%I

%S 1,3,20,218,3414,70306,1789850,54071216,1886496960,74588759664,

%T 3295393803888,160898970043632,8603780292835896,500078481148348176,

%U 31391957137745933088,2116613399519305596384,152558384742741641353056,11705479592386152200155200

%N E.g.f. A(x) satisfies: exp(A(x)) = x + 2*exp(A(x)^2) - exp(A(x)^3), with A(0) = 0.

%F E.g.f.: A(x) = Series_Reversion( exp(x) - 2*exp(x^2) + exp(x^3) ).

%F a(n) ~ n^(n-1) / (sqrt(-exp(s) + 4*exp(s^2)*(1+2*s^2) - 3*exp(s^3)*s*(2+3*s^3)) * exp(n) * r^(n-1/2)), where r = 0.21502388560220862525... and s = 0.46516827378128071788 are the roots of the system of equations exp(s) + 3*exp(s^3)*s^2 = 4*exp(s^2)*s, r = exp(s) - 2*exp(s^2) + exp(s^3). - _Vaclav Kotesovec_, Jan 12 2014

%e E.g.f.: A(x) = x + 3*x^2/2! + 20*x^3/3! + 218*x^4/4! + 3414*x^5/5! +...

%e where A( exp(x) - 2*exp(x^2) + exp(x^3) ) = x.

%e Related expansions:

%e exp(A(x)) = 1 + x + 4*x^2/2! + 30*x^3/3! + 344*x^4/4! + 5470*x^5/5! +...

%e 2*exp(A(x)^2) = 2 + 4*x^2/2! + 36*x^3/3! + 452*x^4/4! + 7480*x^5/5! +...

%e exp(A(x)^3) = 1 + 6*x^3/3! + 108*x^4/4! + 2010*x^5/5! +...

%t Rest[CoefficientList[InverseSeries[Series[Exp[x] - 2*Exp[x^2] + Exp[x^3], {x, 0, 20}], x],x] * Range[0, 20]!] (* _Vaclav Kotesovec_, Jan 12 2014 *)

%o (PARI) {a(n)=local(X=x+x*O(x^n));if(n<1, 0, n!*polcoeff(serreverse(exp(X)-2*exp(X^2)+exp(X^3)), n))}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A138014, A206401, A206402, A206403, A206404.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Feb 07 2012

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Last modified January 25 08:48 EST 2022. Contains 350565 sequences. (Running on oeis4.)