OFFSET
1,2
COMMENTS
Note that a(30) is negative. - Vaclav Kotesovec, Sep 16 2013
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..370
Vaclav Kotesovec, Graph of convergence to limit for 1000 terms
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
EXAMPLE
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) +...
MATHEMATICA
Rest[CoefficientList[InverseSeries[Series[x-1+E^(-x^2), {x, 0, 20}], x], x]*Range[0, 20]!] (* Vaclav Kotesovec, Sep 16 2013 *)
PROG
(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), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Aug 14 2013
STATUS
approved