

A228180


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


2



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
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OFFSET

0,3


COMMENTS

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


REFERENCES

D. E. Davenport, L. K. Pudwell, L. W. Shapiro, L. C. Woodson, The Boundary of Ordered Trees, 2014; http://faculty.valpo.edu/lpudwell/papers/treeboundary.pdf


LINKS

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


FORMULA

G.f.: (x*C+2*x^3*C^4)/(1x) where C is the g.f. for the Catalan numbers A000108.
Conjecture: 2*(n+1)*a(n) +(13*n+5)*a(n1) +(23*n37)*a(n2) +6*(2*n+5)*a(n3)=0.  R. J. Mathar, Aug 25 2013
a(n) ~ 5*4^n / (3*sqrt(Pi)*n^(3/2)).  Vaclav Kotesovec, Feb 01 2014


EXAMPLE

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.


MATHEMATICA

CoefficientList[Series[(x*(1Sqrt[14*x])/(2*x) + 2*x^3*((1Sqrt[14*x])/(2*x))^4)/(1x), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 01 2014 *)


PROG

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


CROSSREFS

Cf. A000108, A228178.
Sequence in context: A138747 A052975 A275943 * A035929 A071646 A114627
Adjacent sequences: A228177 A228178 A228179 * A228181 A228182 A228183


KEYWORD

nonn


AUTHOR

Louis Shapiro, Aug 20 2013


STATUS

approved



