
COMMENTS

Number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0, s(1) = 1 = s(n), s(i)  s(i1) <= 1 for i >= 2. Also a(n) = T(n,n1), where T is array in A026105 and U(n,n+1), where U is array in A026120.
Also number of (s(0),s(1),...,s(n)) such that every s(i) is a nonnegative integer, s(0) = 1, s(n) = 0, s(1)  s(0) = 1, s(i)  s(i1) <= 1 for i >= 2.
Number of Motzkin paths of length n+1 that start with a (1,1) step and end with a (1,1) step.  Emeric Deutsch, Jul 11 2001
The sequence 1,1,3,7,18.... has a(n)=sum{k=0..n, C(n,2k)*A000108(k+1) }.  Paul Barry, Jul 18 2003
Equals iterates of M * [1,1,1,1,0,0,0,...] where M = an infinite tridiagonal matrix with [0,1,1,1,...] in the main diagonal and [1,1,1,...] in the super and subdiagonals. [From Gary W. Adamson, Jan 08 2009]
Motzkin paths of length n1 that are allowed to go down to the line y=1 [HeShapiro, page 38].  R. J. Mathar, Jul 23 2017
