A027302 a(n) = Sum_{k=0..floor((n-1)/2)} T(n,k) * T(n,k+1), with T given by A008315.

1, 2, 9, 24, 95, 286, 1099, 3536, 13479, 45220, 172150, 594320, 2265003, 7983990, 30487175, 109174560, 417812417, 1514797020, 5810065898, 21275014800, 81775140083, 301892460012, 1162703549474, 4321730134624, 16675372590850, 62340424959176, 240949471232124

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

Table of n, a(n) for n=1..27.

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

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

Sequence in context: A200086 A143561 A363660 * A213720 A353822 A073981

Adjacent sequences: A027299 A027300 A027301 * A027303 A027304 A027305

Keyword

nonn

Author

Clark Kimberling

Extensions

More terms from Sean A. Irvine, Oct 26 2019

Status

approved