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A027302
a(n) = Sum_{k=0..floor((n-1)/2)} T(n,k) * T(n,k+1), with T given by A008315.
2
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
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
Sequence in context: A200086 A143561 A363660 * A374703 A213720 A353822
KEYWORD
nonn,changed
EXTENSIONS
More terms from Sean A. Irvine, Oct 26 2019
STATUS
approved