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

1,2

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

Cf. A218652.

Sequence in context: A210997 A052563 A316704 * A218652 A194786 A339301

Adjacent sequences: A228170 A228171 A228172 * A228174 A228175 A228176

Keyword

sign

Author

Paul D. Hanna, Aug 14 2013

Status

approved