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A206404 E.g.f. A(x) satisfies: exp(-A(x)) = exp(-A(x)^2) - x, with A(0) = 0. 4

%I

%S 1,3,26,364,7074,176108,5348132,191725840,7924856460,371061933552,

%T 19411323110904,1122067341369984,71024428188382200,

%U 4885895673623299008,362955565203398550768,28957167717593649778176,2469386593299982674585744,224154175905500071395278592

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

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

%F a(n) ~ sqrt((1-2*s)/(2+2*s-4*s^2)) * n^(n-1) / (exp(1-s^2)-exp(1-s))^n, where s = 0.393815762008795197237... is the root of the equation exp(s^2) = 2*s*exp(s). - _Vaclav Kotesovec_, Jan 12 2014

%e E.g.f.: A(x) = x + 3*x^2/2! + 26*x^3/3! + 364*x^4/4! + 7074*x^5/5! +...

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

%e Related expansions:

%e exp(-A(x)) = 1 - x - 2*x^2/2! - 18*x^3/3! - 250*x^4/4! - 4840*x^5/5! +...

%e exp(-A(x)^2) = 1 - 2*x^2/2! - 18*x^3/3! - 250*x^4/4! - 4840*x^5/5! +...

%t Rest[CoefficientList[InverseSeries[Series[Exp[-x^2] - Exp[-x], {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)), n))}

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

%Y Cf. A138014, A206401, A206402, A206403, A206405.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Feb 07 2012

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Last modified January 26 23:54 EST 2022. Contains 350601 sequences. (Running on oeis4.)