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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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified December 8 13:24 EST 2023. Contains 367679 sequences. (Running on oeis4.)