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A206405
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E.g.f. A(x) satisfies: exp(A(x)) = x + 2*exp(A(x)^2) - exp(A(x)^3), with A(0) = 0.
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4
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1, 3, 20, 218, 3414, 70306, 1789850, 54071216, 1886496960, 74588759664, 3295393803888, 160898970043632, 8603780292835896, 500078481148348176, 31391957137745933088, 2116613399519305596384, 152558384742741641353056, 11705479592386152200155200
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OFFSET
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1,2
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LINKS
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FORMULA
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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
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EXAMPLE
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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! +...
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MATHEMATICA
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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 *)
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PROG
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(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), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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