A228173 E.g.f. satisfies: A(x) = 1+x - exp(-A(x)^2).
1, 2, 12, 108, 1320, 20280, 374640, 8072400, 198465120, 5475284640, 167285321280, 5600184004800, 203602252613760, 7978382871338880, 334767145102790400, 14952953514231532800, 707221717016278464000, 35242469168705967168000, 1841491290250262851200000
Offset: 1
Keywords
Examples
E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 108*x^4/4! + 1320*x^5/5! +... where exp(-A(x)^2) = 1 - 2*x^2/2! - 12*x^3/3! - 108*x^4/4! - 1320*x^5/5! -... The e.g.f. equals the series: 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! +... Also, 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) +...
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 1..370
- Vaclav Kotesovec, Graph of convergence to limit for 1000 terms
Crossrefs
Cf. A218652.
Programs
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Mathematica
Rest[CoefficientList[InverseSeries[Series[x-1+E^(-x^2),{x,0,20}],x],x]*Range[0,20]!] (* Vaclav Kotesovec, Sep 16 2013 *) -
PARI
{a(n)=n!*polcoeff(serreverse(x-1+exp(-x^2+x*O(x^n))), n)} for(n=1, 25, print1(a(n), ", ")) -
PARI
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} {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)} for(n=1, 25, print1(a(n), ", ")) -
PARI
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} {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)} for(n=1, 25, print1(a(n), ", "))
Formula
E.g.f. A(x) satisfies:
(1) A(x - 1 + exp(-x^2)) = x.
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) (1 - exp(-x^2))^n / n!.
(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (1 - exp(-x^2))^n/x / n! ).
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
Comments