OFFSET
1,2
COMMENTS
a(n) is the number of Dyck (n+2)-paths with UU spanning the midpoint. E.g., for n=2 the two Dyck 4-paths are UUDU.UDDD and UDUU.UDDD where dot marks the midpoint. - David Scambler, Feb 11 2011
Apparently also the number of returns to the left of or to the midpoint of all Dyck paths with semilength n+1. - David Scambler, Apr 30 2013
LINKS
Alon Regev, The central component of a triangulation, arXiv:1210.3349 [math.CO], 2012, see p. 6.
Alon Regev, The Central Component of a Triangulation, J. Int. Seq. 16 (2013) #13.4.1
FORMULA
Conjecture D-finite with recurrence -(n+2)*(13*n-2)*(3+n)^2*a(n) +10*(8*n^2+3*n-8)*(n+2)^2*a(n-1) +8*(12*n^4+47*n^3+52*n^2+67*n+20)*a(n-2) -160*(8*n^2+3*n-8)*(n-1)^2*a(n-3) +128*(7*n+4)*(2*n-5)*(-2+n)^2*a(n-4)=0. - R. J. Mathar, Nov 22 2024
MATHEMATICA
a[n_] := With[{C = CatalanNumber}, Sum[C[k]*C[n+1-k], {k, 1, (n+1)/2}]]; Array[a, 30] (* Jean-François Alcover, May 01 2017 *)
PROG
(Sage)
def C(n): return binomial(2*n, n)/(n+1) # Catalan numbers
def A027302(n): return add(C(k)*C(n+1-k) for k in (1..(n+1)/2))
[A027302(n) for n in (1..22)] # Peter Luschny, Jun 27 2013
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
EXTENSIONS
More terms from Sean A. Irvine, Oct 26 2019
STATUS
approved