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 A228173 E.g.f. satisfies: A(x) = 1+x - exp(-A(x)^2). 1

%I

%S 1,2,12,108,1320,20280,374640,8072400,198465120,5475284640,

%T 167285321280,5600184004800,203602252613760,7978382871338880,

%U 334767145102790400,14952953514231532800,707221717016278464000,35242469168705967168000,1841491290250262851200000

%N E.g.f. satisfies: A(x) = 1+x - exp(-A(x)^2).

%C Note that a(30) is negative. - _Vaclav Kotesovec_, Sep 16 2013

%H Vaclav Kotesovec, <a href="/A228173/b228173.txt">Table of n, a(n) for n = 1..370</a>

%H Vaclav Kotesovec, <a href="/A228173/a228173.jpg">Graph of convergence to limit for 1000 terms</a>

%F E.g.f. A(x) satisfies:

%F (1) A(x - 1 + exp(-x^2)) = x.

%F (2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) (1 - exp(-x^2))^n / n!.

%F (3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (1 - exp(-x^2))^n/x / n! ).

%F Lim sup n->infinity (|a(n)|/n!)^(1/n) = 1/abs(-1-(LambertW(-1/2)-1) / sqrt(-2*LambertW(-1/2))) = 3.19002880735268... - _Vaclav Kotesovec_, Jan 11 2014

%e E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 108*x^4/4! + 1320*x^5/5! +...

%e where

%e exp(-A(x)^2) = 1 - 2*x^2/2! - 12*x^3/3! - 108*x^4/4! - 1320*x^5/5! -...

%e The e.g.f. equals the series:

%e A(x) = x + (1 - exp(-x^2)) + d/dx (1 - exp(-x^2))^2/2! + d^2/dx^2 (1 - exp(-x^2))^3/3! + d^3/dx^3 (1 - exp(-x^2))^4/4! + d^4/dx^4 (1 - exp(-x^2))^5/5! +...

%e Also,

%e log(A(x)/x) = (1 - exp(-x^2))/x + d/dx (1 - exp(-x^2))^2/(2!*x) + d^2/dx^2 (1 - exp(-x^2))^3/(3!*x) + d^3/dx^3 (1 - exp(-x^2))^4/(4!*x) +...

%t Rest[CoefficientList[InverseSeries[Series[x-1+E^(-x^2),{x,0,20}],x],x]*Range[0,20]!] (* _Vaclav Kotesovec_, Sep 16 2013 *)

%o (PARI) {a(n)=n!*polcoeff(serreverse(x-1+exp(-x^2+x*O(x^n))), n)}

%o for(n=1, 25, print1(a(n), ", "))

%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

%o {a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, (1 - exp(-x^2+x*O(x^n)))^m)/m!); n!*polcoeff(A, n)}

%o for(n=1, 25, print1(a(n), ", "))

%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

%o {a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, (1 - exp(-x^2+x*O(x^n)))^m/x)/m!)+x*O(x^n)); n!*polcoeff(A, n)}

%o for(n=1, 25, print1(a(n), ", "))

%Y Cf. A218652.

%K sign

%O 1,2

%A _Paul D. Hanna_, Aug 14 2013

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Last modified December 8 14:38 EST 2019. Contains 329865 sequences. (Running on oeis4.)