A213093 G.f. satisfies: A(x) = 1 + x/A(-x*A(x)^4).

1, 1, 1, 4, 13, 62, 297, 1523, 8091, 43243, 234347, 1267141, 6814076, 36368431, 192079140, 1006805203, 5262612068, 27656507707, 147973596219, 815825605806, 4662818005761, 27504894986209, 165036600363916, 989160502170958, 5829789341752240, 33444482725193880

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(42) = -16825305705383790675462237694. - Georg Fischer, Feb 16 2019

Examples

			G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 13*x^4 + 62*x^5 + 297*x^6 + 1523*x^7 +...
Related expansions:
A(x)^4 = 1 + 4*x + 10*x^2 + 32*x^3 + 119*x^4 + 516*x^5 + 2462*x^6 +...
A(-x*A(x)^4) = 1 - x - 3*x^2 - 6*x^3 - 31*x^4 - 141*x^5 - 697*x^6 - 3641*x^7 -...
		

Links

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

Crossrefs

Cf. A000108, A001764, A213091, A213092, A213094, A213095, A213096.

Programs

Mathematica

nmax = 25; 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]) + 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^4,x,x+x*O(x^n))) );polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))