OFFSET
1,2
COMMENTS
Compare to: W'(x) = exp(W(x)) / (1 - W(x)), which holds when W(x) = -LambertW(-x).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..330
V. Kotesovec, Asymptotic of implicit functions if Fww = 0.
FORMULA
E.g.f.: Series_Reversion( Sum_{n>=1} -(n^2-n+1) * (-x)^n/n! ).
a(n) ~ Gamma(1/3) * n^(n-5/6) / (3^(1/6) * sqrt(2*Pi) * (exp(1)-2)^(n-1/3)). - Vaclav Kotesovec, Apr 26 2015
EXAMPLE
E.g.f.: A(x) = x + 3*x^2/2! + 20*x^3/3! + 208*x^4/4! + 2944*x^5/5! + ...
where
exp(A(x))/(1-A(x))^2 = 1 + 3*x + 20*x^2/2! + 208*x^3/3! + 2944*x^4/4! + ...
The e.g.f. also satisfies:
x = Sum_{n>=1} (-1)^(n-1) * (n^2-n+1) * A(x)^n/n!, or
x = 1 - exp(-A(x)) * (1 + A(x)^2) [after Vaclav Kotesovec];
explicitly,
x = A(x)*x - 3*A(x)^2*x^2/2! + 7*A(x)^3*x^3/3! - 13*A(x)^4*x^4/4! + 21*A(x)^5*x^5/5! - 31*A(x)^6*x^6/6! + ...
MATHEMATICA
Rest[CoefficientList[InverseSeries[Series[1-E^(-x)*(1+x^2), {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Apr 26 2015 *)
PROG
(PARI) {a(n)=local(A=x); for(i=1, n, A = intformal((1-A)^-2*(exp(A+x*O(x^n)))) ); n!*polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=x); A = serreverse(sum(m=1, n, (-1)^(m-1)*(m^2-m+1)*x^m/m!) +x*O(x^n)); n!*polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 25 2015
STATUS
approved