
COMMENTS

Number of protected vertices in all ordered rooted trees with n edges. A protected vertex in an ordered tree is a vertex at least 2 edges away from its leaf descendants.  Emeric Deutsch, Aug 20 2008
1,3,11,... gives the diagonal sums of A111418. Hankel transform of a(n) is A128834. Hankel transform of a(n+1) is A187340.  Paul Barry, Mar 08 2011
a(n) = A035317(2*n1,n1) for n > 0.  Reinhard Zumkeller, Jul 19 2012
Apparently the number of peaks in all Dyck paths of semilength n+1 that are the same height as the preceding peak.  David Scambler, Apr 22 2013
Define an infinite triangle by T(n,0)=A001045(n) (the first column) and T(r,c) = sum_{k=c1, c,..,r} T(k,c1), (the sum of all the terms in the preceding column down to row r. Then T(n,n)=a(n+1). The triangle is 0; 1,1; 1,2,3; 3,5,8,11; 5,10,18,29,40; 11,21,39,68,108,148;... Example: T(5,2)=39=the sum of the terms in column 1 from T(1,1) to T(5,1), namely, 1+2+5+10+21.  J. M. Bergot, May 17 2013
Also for n>=1 the number of unimodal functions f:[n]>[n] with f(1)<>1 and f(i)<>f(i+1). a(4) = 11: [2,3,2,1], [2,3,4,1], [2,3,4,2], [2,3,4,3], [2,4,2,1], [2,4,3,1], [2,4,3,2], [3,4,2,1], [3,4,3,1], [3,4,3,2], [4,3,2,1].  Alois P. Heinz, May 23 2013
