A213094 G.f. satisfies: A(x) = 1 + x/A(-x*A(x)^4)^2.

1, 1, 2, 7, 26, 123, 622, 3490, 20468, 125643, 792606, 5118050, 33612998, 223770400, 1505528080, 10213807498, 69746716716, 478693572991, 3298184837434, 22790090901504, 157803590073220, 1094189186549354, 7593267782966708, 52713912426435111, 365948276764762712

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

Cf. A000108, A001764, A213091, A213092, A213093, A213095, A213096.

Sequence in context: A218670 A302691 A081566 * A141203 A346749 A096803

Adjacent sequences: A213091 A213092 A213093 * A213095 A213096 A213097

Keyword

sign

Author

Paul D. Hanna, Jun 05 2012

Status

approved