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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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Apparently the partial sums of A070031. - R. J. Mathar, Aug 25 2013
LINKS
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.
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
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*(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 *)
PROG
(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
CROSSREFS
Sequence in context: A138747 A052975 A275943 * A035929 A071646 A114627
KEYWORD
nonn
AUTHOR
Louis Shapiro, Aug 20 2013
STATUS
approved

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Last modified March 19 06:32 EDT 2024. Contains 370953 sequences. (Running on oeis4.)