%I A064645
%S A064645 1,1,1,2,1,1,4,1,1,1,9,2,1,1,1,21,4,1,1,1,1,51,8,2,1,1,1,1,127,17,4,1,
               1,1,1,1,323,37,8,2,
%T A064645 1,1,1,1,1,835,82,16,4,1,1,1,1,1,1,2188,185,33,8,2,1,1,1,1,1,1,5798,423,
               69,16,4,1,1,1,1,
%U A064645 1,1,1,15511,978,146,32,8,2,1,1,1,1,1,1,1,41835,2283,312,65,16,4,1,1,1,
               1,1,1,1,1
%N A064645 Table where the entry (n,k) (n >= 0,k >= 0) gives number of Motzkin paths 
               of the length n with the minimum peak width of k.
%e A064645 E.g. we have the following nine Motzkin paths of length 4, of which 4 
               last have all peaks at least of width 1 and two last with any peak 
               at least 2 dashes wide, so M(4,0) = 9, M(4,1) = 4 and M(4,2) = 2.
%e A064645 ./\............................_......_....__
%e A064645 /..\../\/\..__/\.._/\_../\__../.\_.._/.\../..\..____
%p A064645 [seq(A064645(j),j=0..104)]; A064645 := (n) -> Mpw((((trinv(n)-1)*(((1/
               2)*trinv(n))+1))-n), (n-((trinv(n)*(trinv(n)-1))/2)));
%p A064645 C := (n,k) -> `if`((n <= 0),0,binomial(n,k));
%p A064645 Mpw := proc(n,m) local i,k; 1+add(add(A001263(i,k)*C(n-(m*k),2*i),k=1..i),
               i=0..floor(n/2)); end;
%Y A064645 First row (k=0): Motzkin numbers (A001006), second row (k=1): A004148 
               (with RNA connection), third row (k=2): A004149, fourth row (k=3): 
               A023421, fifth row (k=4): A023422, sixth row (k=5): A023423. Uses 
               the table A001263(n, k) which gives the Dyck paths (Catalan Mountain 
               Ranges) with exactly k peaks. Maple procedure trinv given at A054425.
%Y A064645 Sequence in context: A106246 A136674 A144383 this_sequence A008307 A099238 
               A141450
%Y A064645 Adjacent sequences: A064642 A064643 A064644 this_sequence A064646 A064647 
               A064648
%K A064645 nonn,tabl
%O A064645 0,4
%A A064645 Antti Karttunen Oct 03 2001