A228180 The number of single edges on the boundary of ordered trees with n edges.

0, 1, 2, 6, 19, 61, 199, 661, 2234, 7668, 26674, 93858, 333524, 1195288, 4315468, 15681838, 57312643, 210529213, 776872243, 2878482523, 10704933793, 39945106573, 149511432793, 561182969173, 2111812422871, 7965992783803, 30114859723751, 114079902339303, 432975153092011, 1646215731143667

Offset: 0

Author

Louis Shapiro, Aug 20 2013

Keywords

• nonn

Comments

Apparently the partial sums of A070031. - R. J. Mathar, Aug 25 2013

Examples

			For n=3 the UUUDDD has 3 single edges while UUDDUD, UDUUDD and UUDUDD each have one single edge, i.e., an edge with no siblings.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Dennis E. Davenport, Lara K. Pudwell, Louis W. Shapiro, and Leon C. Woodson, The Boundary of Ordered Trees, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.8; preprint, 2014.
• W. Kuszmaul, Fast Algorithms for Finding Pattern Avoiders and Counting Pattern Occurrences in Permutations, arXiv preprint arXiv:1509.08216 [cs.DM], 2015-2017.

Crossrefs

Cf. A000108, A228178.

Programs

Mathematica

CoefficientList[Series[(x*(1-Sqrt[1-4*x])/(2*x) + 2*x^3*((1-Sqrt[1-4*x])/(2*x))^4)/(1-x), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 01 2014 *)

PARI

x = 'x + O('x^66);
C = serreverse( x/( 1/(1-x) ) ) / x; \\ Catalan A000108
gf = (x*C+2*x^3*C^4)/(1-x);
concat([0], Vec(gf) ) \\ Joerg Arndt, Aug 21 2013

Formula

G.f.: (x*C+2*x^3*C^4)/(1-x) where C is the g.f. for the Catalan numbers A000108.

Conjecture: 2*(n+1)*a(n) +(-13*n+5)*a(n-1) +(23*n-37)*a(n-2) +6*(-2*n+5)*a(n-3)=0. - R. J. Mathar, Aug 25 2013

a(n) ~ 5*4^n / (3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Feb 01 2014