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%I #37 Aug 01 2022 00:07:26
%S 0,1,2,6,19,61,199,661,2234,7668,26674,93858,333524,1195288,4315468,
%T 15681838,57312643,210529213,776872243,2878482523,10704933793,
%U 39945106573,149511432793,561182969173,2111812422871,7965992783803,30114859723751,114079902339303,432975153092011,1646215731143667
%N The number of single edges on the boundary of ordered trees with n edges.
%C Apparently the partial sums of A070031. - _R. J. Mathar_, Aug 25 2013
%H Vincenzo Librandi, <a href="/A228180/b228180.txt">Table of n, a(n) for n = 0..200</a>
%H Dennis E. Davenport, Lara K. Pudwell, Louis W. Shapiro, and Leon C. Woodson, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Davenport/dav3.html">The Boundary of Ordered Trees</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.8; <a href="http://faculty.valpo.edu/lpudwell/papers/treeboundary.pdf">preprint</a>, 2014.
%H W. Kuszmaul, <a href="http://arxiv.org/abs/1509.08216">Fast Algorithms for Finding Pattern Avoiders and Counting Pattern Occurrences in Permutations</a>, arXiv preprint arXiv:1509.08216 [cs.DM], 2015-2017.
%F G.f.: (x*C+2*x^3*C^4)/(1-x) where C is the g.f. for the Catalan numbers A000108.
%F 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
%F a(n) ~ 5*4^n / (3*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Feb 01 2014
%e 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.
%t 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 *)
%o (PARI)
%o x = 'x + O('x^66);
%o C = serreverse( x/( 1/(1-x) ) ) / x; \\ Catalan A000108
%o gf = (x*C+2*x^3*C^4)/(1-x);
%o concat([0], Vec(gf) ) \\ _Joerg Arndt_, Aug 21 2013
%Y Cf. A000108, A228178.
%K nonn
%O 0,3
%A _Louis Shapiro_, Aug 20 2013
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