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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
1, 3, 20, 218, 3414, 70306, 1789850, 54071216, 1886496960, 74588759664, 3295393803888, 160898970043632, 8603780292835896, 500078481148348176, 31391957137745933088, 2116613399519305596384, 152558384742741641353056, 11705479592386152200155200 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Table of n, a(n) for n=1..18.

FORMULA

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

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

EXAMPLE

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

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

Related expansions:

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

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

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

MATHEMATICA

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 *)

PROG

(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))}

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

CROSSREFS

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

Sequence in context: A006963 A243426 A113333 * A307363 A052851 A262233

Adjacent sequences:  A206402 A206403 A206404 * A206406 A206407 A206408

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 07 2012

STATUS

approved

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Last modified December 8 07:37 EST 2021. Contains 349593 sequences. (Running on oeis4.)