A186031 Number of Dyck paths of semilength n with a valley (DU) spanning the midpoint.

0, 1, 1, 5, 10, 42, 107, 429, 1234, 4862, 15032, 58786, 190588, 742900, 2490399, 9694845, 33312770, 129644790, 453999656, 1767263190, 6282014804, 24466267020, 88026769844, 343059613650, 1246635155180, 4861946401452, 17815452662152

Offset

1,4

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Formula

a(n:even) = A000108(n-1); a(n:odd) = A000108(n-1) - A000108((n-1)/2)^2.

Example

For n=4 the 5 paths are UDUD.UDUD, UUDD.UDUD, UDUD.UUDD, UUDD.UUDD and UUUD.UDDD where dot marks the midpoint.

Maple

C:= n-> binomial (2*n, n)/(n+1):
a:= n-> C(n-1) -`if` (irem(n, 2)=1, C((n-1)/2)^2, 0):
seq (a(n), n=1..40);

Crossrefs

Dyck n-paths with peak at midpoint is given by A000108(n-1); For UU (or DD) at midpoint see A027302.

Sequence in context: A270077 A271150 A271202 * A305246 A316546 A187877

Adjacent sequences: A186028 A186029 A186030 * A186032 A186033 A186034

Keyword

nonn

Author

David Scambler, Feb 11 2011

Status

approved