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Search: id:A097887
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| A097887 |
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Triangle read by rows: T(n,k) is the number of Motzkin paths of length n having k low humps. (A hump is an upstep followed by 0 or more flatsteps followed by a downstep. A low hump is a hump that starts at level zero.). |
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+0
1
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| 1, 1, 1, 1, 1, 3, 2, 6, 1, 6, 10, 5, 18, 17, 15, 1, 50, 35, 35, 7, 133, 88, 73, 28, 1, 349, 240, 153, 84, 9, 919, 658, 351, 214, 45, 1, 2443, 1782, 891, 506, 165, 11, 6559, 4792, 2397, 1196, 500, 66, 1, 17759, 12896, 6565, 2964, 1352, 286, 13, 48417, 34892, 17993, 7765
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OFFSET
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0,6
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COMMENT
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Row sums are the Motzkin numbers (A001006). Column 0 gives A089380.
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FORMULA
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G.f.=G=G(t, z) satisfies z^2*(2-4z+3z^2-t+2tz-3tz^2+t^2*z^2)G^2-(1-z)(1-2z+3z^2-2tz^2)G+(1-z)^2=0.
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EXAMPLE
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Triangle begins:
1;
1;
1,1;
1,3;
2,6,1;
6,10,5;
18,17,15,1;
Row n contains 1+floor(n/2) terms.
T(5,2)=5 counts (UD)(UHD), (UHD)(UD), H(UD)(UD), (UD)H(UD) and (UD)(UD)H, where U=(1,1), H=(1,0), D=(1,-1) (the low humps are shown between parentheses).
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CROSSREFS
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Cf. A001006, A089380.
Sequence in context: A005266 A005267 A016460 this_sequence A019761 A133519 A135223
Adjacent sequences: A097884 A097885 A097886 this_sequence A097888 A097889 A097890
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 02 2004
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