OFFSET
0,3
COMMENTS
Compare definition of g.f. to:
(1) B(x) = 1 + x/B(-x*B(x)) when B(x) = 1/(1-x).
(2) C(x) = 1 + x/C(-x*C(x)^3)^2 when C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
(3) D(x) = 1 + x/D(-x*D(x)^5)^3 when D(x) = 1 + x*D(x)^3 is the g.f. of the ternary tree numbers (A001764).
The first negative term is a(136). - Georg Fischer, Feb 16 2019
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 26*x^4 + 123*x^5 + 622*x^6 + 3490*x^7 +...
Related expansions:
A(x)^4 = 1 + 4*x + 14*x^2 + 56*x^3 + 237*x^4 + 1112*x^5 + 5614*x^6 +...
A(-x*A(x)^4)^2 = 1 - 2*x - 3*x^2 - 6*x^3 - 38*x^4 - 180*x^5 - 1095*x^6 -...
MATHEMATICA
nmax = 24; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x] - (1 + x/A[-x A[x]^4]^2) + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. Rule -> Set;
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 01 2019 *)
PROG
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+x/subst(A^2, x, -x*subst(A^4, x, x+x*O(x^n))) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jun 05 2012
STATUS
approved