Sequence counting question

[franktaw at netscape.net]

Consider finite sequences of positive integers <b(m)> of length n with b(1)=1, and with the constraint that b(m) <= max_{0<k<n} b(k)+k-n+2.  The question is how many such sequences there are.
 
(Note that when we consider only the term k=m-1, this becomes b(m) <= b(m-1)+1, and it is well known that the number of sequences under this constraint is the Catalan numbers.)
 
This sequence starts (from n = 1) 1,2,5,14,43,142,499,1850,7193.
 
This appears to be A005425 (this many terms matching is not going to be a coincidence).  But I don't see any way to prove it.
 
Franklin T. Adams-Watters
16 W. Michigan Ave.
Palatine, IL 60067
847-776-7645
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