A186031 - OEIS

%I #9 Nov 02 2013 22:39:53

%S 0,1,1,5,10,42,107,429,1234,4862,15032,58786,190588,742900,2490399,

%T 9694845,33312770,129644790,453999656,1767263190,6282014804,

%U 24466267020,88026769844,343059613650,1246635155180,4861946401452,17815452662152

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

%H Alois P. Heinz, <a href="/A186031/b186031.txt">Table of n, a(n) for n = 1..1000</a>

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

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

%p C:= n-> binomial (2*n, n)/(n+1):

%p a:= n-> C(n-1) -`if` (irem(n, 2)=1, C((n-1)/2)^2, 0):

%p seq (a(n), n=1..40);

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

%K nonn

%O 1,4

%A _David Scambler_, Feb 11 2011