A071879 G.f. satisfies: A(x) = 1 + x*A(x) + x^3*A(x)^3.

1, 1, 1, 2, 5, 11, 24, 57, 141, 349, 871, 2212, 5688, 14730, 38403, 100829, 266333, 706997, 1885165, 5047522, 13565203, 36578497, 98934826, 268342933, 729709432, 1989021256, 5433518806, 14873285506, 40790118487, 112064912455

Offset

0,4

Comments

Number of ordered trees with n edges and having nonleaf nodes of outdegrees 1 or 3. - Emeric Deutsch, Nov 03 2002. [Comment corrected by Christian G. Bower, Sep 25 2007]

Sequence is a Motzkin-like sequence. The Motzkin sequence A001006 counts ordered trees with n edges and having nodes of outdegree 0, 1, or 2 [g.f. f(x) defined by f = 1+x*f+(x*f)^2]. - Emeric Deutsch, Sep 30 2007

G.f. (offset 1) is series reversion of x^2/(x+x^2+x^4).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.

Olivier Bodini, Matthieu Dien, Antoine Genitrini, and Frédéric Peschanski, The Ordered and Colored Products in Analytic Combinatorics: Application to the Quantitative Study of Synchronizations in Concurrent Processes. In 2017 Proceedings of the Fourteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO).

Isaac DeJager, Madeleine Naquin, and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.

Formula

a(n) = (Sum_{i=0..floor(n/3)} C(n+1, 1+2i)*C(n-2i, i))/(n+1). - Emeric Deutsch, Nov 03 2002

a(n) = Sum_{k=0..floor(n/3)} C(n,3k)*C(3k,k)/(2k+1). - Paul Barry, Sep 07 2006

D-finite with recurrence: 2*n*(2*n+3)*a(n) + 2*(1-6*n^2)*a(n-1) + 6*(2*n-1)*(n-1)*a(n-2) - 31*(n-1)*(n-2)*a(n-3) = 0. - R. J. Mathar, Nov 13 2012

a(n) ~ (2+3*2^(1/3))^(3/2) * (1+3*2^(-2/3))^n/(4*sqrt(6*Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 19 2013

G.f. satisfies: A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k) * (x*A(x))^(2*k). - Paul D. Hanna, Sep 05 2014

Example

G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 11*x^5 + 24*x^6 + ...
The first-order differences of the terms form the coefficients of A(x)^3:
A(x)^3 = 1 + 3*x + 6*x^2 + 13*x^3 + 33*x^4 + 84*x^5 + 208*x^6 + 522*x^7 + ...

Maple

a:= n-> add(binomial(n+1, 1+2*i)*binomial(n-2*i, i), i=0..floor(n/3))/(n+1): seq(a(n), n=0..29);

Mathematica

a[n_] := Sum[Binomial[n+1, 1+2i]*Binomial[n-2i, i], {i, 0, Floor[n/3]}]/(n+1);

Prog

(PARI) a(n)=if(n<0, 0, polcoeff(serreverse(x^2/(x+x^2+x^4+x^2*O(x^n))), n+1))
(PARI) Vec(serreverse(x/(1+x+x^3)+O(x^66))/x) /* Joerg Arndt, Aug 19 2012 */
(PARI) {a(n)=local(A=1); for(i=1, n, A=sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)*(x*A)^(2*k)) +x*O(x^n))); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", ")) \\ Paul D. Hanna, Sep 05 2014

Crossrefs

Cf. A001006, A001764, A116411.

Sequence in context: A242551 A018007 A295091 * A228862 A134527 A124379

Adjacent sequences: A071876 A071877 A071878 * A071880 A071881 A071882

Keyword

nonn

Author

Paul D. Hanna, Jun 10 2002

Extensions

Name changed by Paul D. Hanna, Nov 15 2012

Status

approved