This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A186031 #9 Nov 02 2013 22:39:53 %S A186031 0,1,1,5,10,42,107,429,1234,4862,15032,58786,190588,742900,2490399, %T A186031 9694845,33312770,129644790,453999656,1767263190,6282014804, %U A186031 24466267020,88026769844,343059613650,1246635155180,4861946401452,17815452662152 %N A186031 Number of Dyck paths of semilength n with a valley (DU) spanning the midpoint. %H A186031 Alois P. Heinz, <a href="/A186031/b186031.txt">Table of n, a(n) for n = 1..1000</a> %F A186031 a(n:even) = A000108(n-1); a(n:odd) = A000108(n-1) - A000108((n-1)/2)^2. %e A186031 For n=4 the 5 paths are UDUD.UDUD, UUDD.UDUD, UDUD.UUDD, UUDD.UUDD and UUUD.UDDD where dot marks the midpoint. %p A186031 C:= n-> binomial (2*n, n)/(n+1): %p A186031 a:= n-> C(n-1) -`if` (irem(n, 2)=1, C((n-1)/2)^2, 0): %p A186031 seq (a(n), n=1..40); %Y A186031 Dyck n-paths with peak at midpoint is given by A000108(n-1); For UU (or DD) at midpoint see A027302. %K A186031 nonn %O A186031 1,4 %A A186031 _David Scambler_, Feb 11 2011