A071879 - OEIS

%I #57 Dec 01 2022 15:59:48

%S 1,1,1,2,5,11,24,57,141,349,871,2212,5688,14730,38403,100829,266333,

%T 706997,1885165,5047522,13565203,36578497,98934826,268342933,

%U 729709432,1989021256,5433518806,14873285506,40790118487,112064912455

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

%C 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]

%C 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

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

%H Seiichi Manyama, <a href="/A071879/b071879.txt">Table of n, a(n) for n = 0..1000</a>

%H Paul Barry, <a href="https://doi.org/10.1016/j.laa.2015.10.032">Riordan arrays, generalized Narayana triangles, and series reversion</a>, Linear Algebra and its Applications, 491 (2016) 343-385.

%H Olivier Bodini, Matthieu Dien, Antoine Genitrini, and Frédéric Peschanski, <a href="https://doi.org/10.1137/1.9781611974775.2">The Ordered and Colored Products in Analytic Combinatorics: Application to the Quantitative Study of Synchronizations in Concurrent Processes</a>. In 2017 Proceedings of the Fourteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO).

%H Isaac DeJager, Madeleine Naquin, and Frank Seidl, <a href="https://www.valpo.edu/mathematics-statistics/files/2019/08/Drube2019.pdf">Colored Motzkin Paths of Higher Order</a>, VERUM 2019.

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

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

%F 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

%F 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

%F 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

%e G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 11*x^5 + 24*x^6 + ...

%e The first-order differences of the terms form the coefficients of A(x)^3:

%e A(x)^3 = 1 + 3*x + 6*x^2 + 13*x^3 + 33*x^4 + 84*x^5 + 208*x^6 + 522*x^7 + ...

%p 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);

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

%o (PARI) a(n)=if(n<0,0,polcoeff(serreverse(x^2/(x+x^2+x^4+x^2*O(x^n))),n+1))

%o (PARI) Vec(serreverse(x/(1+x+x^3)+O(x^66))/x) /* _Joerg Arndt_, Aug 19 2012 */

%o (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)}

%o for(n=0, 40, print1(a(n), ", ")) \\ _Paul D. Hanna_, Sep 05 2014

%Y Cf. A001006, A001764, A116411.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Jun 10 2002

%E Name changed by _Paul D. Hanna_, Nov 15 2012