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

Author

Clark Kimberling

Keywords

• nonn

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

Programs

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 *)

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

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

Extensions

More terms from Sean A. Irvine, Oct 26 2019