OFFSET
0,3
COMMENTS
Apparently the partial sums of A070031. - R. J. Mathar, Aug 25 2013
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.
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
KEYWORD
nonn
AUTHOR
Louis Shapiro, Aug 20 2013
STATUS
approved