OFFSET
0,2
COMMENTS
Compare to: W(x) = 1/W(-x*W(x)^2) when W(x) = Sum_{n>=0} (n+1)^(n-1)*x^n/n!.
Compare to: B(x) = 1/B(-x*B(x)^2) when B(x) = Sum_{n>=0} (2*n)!*x^n/n!^2.
An infinite number of functions G(x) satisfy (*) G(x) = 1/G(-x*G(x)^2); for example, (*) is satisfied by G(x) = W(m*x) = LambertW(-m*x)/(-m*x) for all m, where W(x) = Sum_{n>=0} (n+1)^(n-1)*x^n/n!.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..290
FORMULA
The g.f. of this sequence is the limit of the recurrence:
(*) G_{n+1}(x) = (G_n(x) + 1/G_n(-x*G_n(x)^2))/2 starting at G_0(x) = 1+2*x.
EXAMPLE
G.f.: A(x) = 1 + 2*x + 6*x^2 + 24*x^3 + 110*x^4 + 496*x^5 + 2156*x^6 +...
Related expansions:
A(x)^2 = 1 + 4*x + 16*x^2 + 72*x^3 + 352*x^4 + 1720*x^5 + 8192*x^6 +...
1/A(x) = A(-x*A(x)^2) = 1 - 2*x - 2*x^2 - 8*x^3 - 34*x^4 - 112*x^5 - 324*x^6 - 896*x^7 - 1866*x^8 - 800*x^9 + 5540*x^10 +...
PROG
(PARI) {a(n)=local(A=1+2*x); for(i=0, n, A=(A+1/subst(A, x, -x*A^2+x*O(x^n)))/2); polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 29 2012
STATUS
approved