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Search: id:A064645
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| A064645 |
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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. |
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7
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| 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, 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, 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
(list; table; graph; listen)
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OFFSET
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0,4
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EXAMPLE
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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.
./\............................_......_....__
/..\../\/\..__/\.._/\_../\__../.\_.._/.\../..\..____
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MAPLE
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[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)));
C := (n, k) -> `if`((n <= 0), 0, binomial(n, k));
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;
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CROSSREFS
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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.
Sequence in context: A106246 A136674 A144383 this_sequence A008307 A099238 A141450
Adjacent sequences: A064642 A064643 A064644 this_sequence A064646 A064647 A064648
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KEYWORD
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nonn,tabl
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AUTHOR
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Antti Karttunen Oct 03 2001
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