

A027302


a(n) = Sum_{k=0..floor((n1)/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
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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 4paths 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



MATHEMATICA

a[n_] := With[{C = CatalanNumber}, Sum[C[k]*C[n+1k], {k, 1, (n+1)/2}]]; Array[a, 30] (* JeanFrançois 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+1k) for k in (1..(n+1)/2))


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



