OFFSET
0,2
COMMENTS
Compare to: G(x) = 1/G(-x*G(x)^3) when G(x) = 1 + x*G(x)^2 (A000108).
Compare to: B(x) = 1/B(-x*B(x)^3) when B(x) = 1/(1-9*x)^(1/3) = g.f. of A004987.
An infinite number of functions G(x) satisfy (*) G(x) = 1/G(-x*G(x)^3); for example, (*) is satisfied by G(x) = C(m*x) = (1-sqrt(1-4*m*x))/(2*m*x) for all m, where C(x) is the Catalan function.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
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)^3))/2 starting at G_0(x) = 1+2*x.
EXAMPLE
G.f.: A(x) = 1 + 2*x + 8*x^2 + 40*x^3 + 224*x^4 + 1280*x^5 + 7168*x^6 +...
A(x)^2 = 1 + 4*x + 20*x^2 + 112*x^3 + 672*x^4 + 4096*x^5 + 24640*x^6 +...
A(x)^3 = 1 + 6*x + 36*x^2 + 224*x^3 + 1440*x^4 + 9312*x^5 + 59456*x^6 +...
1/A(x) = A(-x*A(x)^3) = 1 - 2*x - 4*x^2 - 16*x^3 - 80*x^4 - 384*x^5 - 1664*x^6 - 7360*x^7 - 40832*x^8 - 304128*x^9 - 2667008*x^10 -...
PROG
(PARI) {a(n)=local(A=1+2*x); for(i=0, n, A=(A+1/subst(A, x, -x*A^3+x*O(x^n)))/2); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 27 2012
STATUS
approved