A168653 G.f. satisfies: A(x*A(x)) = G(x) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

1, 1, 2, 6, 21, 82, 340, 1478, 6622, 30433, 142331, 676203, 3248579, 15776459, 77196573, 380849394, 1888606247, 9430534212, 47236684433, 238214461960, 1202007809362, 6116704517639, 30997312336216, 159384351652358

Offset

0,3

Comments

First negative term is a(32). - Paul D. Hanna, Mar 30 2026

Links

Paul D. Hanna, Table of n, a(n) for n = 0..500

Formula

G.f. satisfies: A(x) = 1 + A(x)^3 * Series_Reversion(x*A(x)).

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

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

Example

G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 21*x^4 + 82*x^5 + 340*x^6 +...
A(x*A(x)) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...

Prog

(PARI) {a(n) = my(A=1+x, F = sum(k=0, n, binomial(3*k+1, k)/(3*k+1)*x^k) + x*O(x^n)); for(i=0, n, A=subst(F, x, serreverse(x*A + x*O(x^n)))); polcoef(GF=A, n)}
{upto(n) = a(n); Vec(GF)}
upto(30)
(PARI) {a(n) = my(A=1+x); for(k=1, n, A = truncate(A);
A = 1 + A^3*serreverse(x*A + x^2*O(x^k)); ); polcoef(GF=A, n)}
{upto(n) = a(n); Vec(GF)} \\ program revised Paul D. Hanna, Mar 30 2026
upto(30)

Crossrefs

Cf. A154677, A168478, A168448, A001764.

Sequence in context: A150218 A150219 A150220 * A279567 A281784 A032347

Adjacent sequences: A168650 A168651 A168652 * A168654 A168655 A168656

Keyword

sign

Author

Paul D. Hanna, Dec 06 2009

Status

approved