A213092 G.f. satisfies: A(x) = 1 + x/A(-x*A(x)^3).

1, 1, 1, 3, 8, 31, 120, 511, 2234, 9988, 45497, 208435, 959496, 4414091, 20252947, 92586100, 421351615, 1910531192, 8647504950, 39194735661, 178643040883, 822295086652, 3836023988259, 18167435295220, 87268076036356, 423657019406289, 2067868784722846

Offset: 0

Author

Paul D. Hanna, Jun 05 2012

Keywords

• sign

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(54) = -4736158402689145255029229896601957. - Georg Fischer, Feb 16 2019

Examples

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 8*x^4 + 31*x^5 + 120*x^6 + 511*x^7 +...
Related expansions:
A(x)^3 = 1 + 3*x + 6*x^2 + 16*x^3 + 48*x^4 + 171*x^5 + 664*x^6 + 2760*x^7 +...
A(-x*A(x)^3) = 1 - x - 2*x^2 - 3*x^3 - 14*x^4 - 50*x^5 - 213*x^6 - 915*x^7 -...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A000108, A001764, A213091, A213093, A213094, A213095, A213096.

Programs

Mathematica

nmax = 26; 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]^3]) + 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 *)

PARI

{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x/subst(A,x,-x*subst(A^3,x,x+x*O(x^n))) );polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))