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 A206403 E.g.f. A(x) satisfies: exp(A(x)) = 2*exp(A(x)^2) - (1-x), with A(0) = 0. 4

%I

%S 1,3,26,398,8604,239296,8135504,326921192,15159790680,796766681184,

%T 46805302872624,3039065898588144,216125148650657232,

%U 16706734205424667296,1394789126514873632832,125073511937467759505760,11989203887017099078716384,1223407961244225521367780096

%N E.g.f. A(x) satisfies: exp(A(x)) = 2*exp(A(x)^2) - (1-x), with A(0) = 0.

%F E.g.f.: A(x) = Series_Reversion( 1 + exp(x) - 2*exp(x^2) ).

%F a(n) ~ sqrt(s/(exp(s)*(1-s+2*s^2))) * n^(n-1) / (exp(n) * (1+exp(s)*(1-1/(2*s)))^(n-1/2)), where s = 0.30957459575023142183... is the root of the equation exp(s) = 4*s*exp(s^2). - _Vaclav Kotesovec_, Jan 12 2014

%e E.g.f.: A(x) = x + 3*x^2/2! + 26*x^3/3! + 398*x^4/4! + 8604*x^5/5! +...

%e where A( 1 + exp(x) - 2*exp(x^2) ) = x.

%e Related expansions:

%e exp(A(x)) = 1 + x + 4*x^2/2! + 36*x^3/3! + 548*x^4/4! + 11800*x^5/5! +...

%e 2*exp(A(x)^2) = 2 + 4*x^2/2! + 36*x^3/3! + 548*x^4/4! + 11800*x^5/5! +...

%t Rest[CoefficientList[InverseSeries[Series[1 + Exp[x] - 2*Exp[x^2], {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(1+exp(X)-2*exp(X^2)), n))}

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

%Y Cf. A138014, A206401, A206402, A206404, A206405.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Feb 07 2012

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Last modified March 21 19:39 EDT 2023. Contains 361410 sequences. (Running on oeis4.)