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A097891
Triangle read by rows: T(n,k) is the number of Motzkin paths of length n and having k peaks at odd height.
0
1, 1, 1, 1, 2, 2, 5, 3, 1, 12, 6, 3, 28, 16, 6, 1, 66, 44, 13, 4, 159, 117, 36, 10, 1, 390, 308, 108, 24, 5, 969, 817, 317, 69, 15, 1, 2432, 2188, 912, 220, 40, 6, 6157, 5898, 2616, 698, 120, 21, 1, 15707, 15968, 7526, 2164, 401, 62, 7, 40340, 43381, 21696, 6638, 1355
OFFSET
0,5
COMMENTS
Row sums are the Motzkin numbers (A001006).
FORMULA
G.f.=G=G(t, z) satisfies z^2*(1-z+z^2-tz^2)G^2-(1-z)(1-z+z^2-tz^2)G+1-z=0.
EXAMPLE
Triangle begins:
1;
1;
1,1;
2,2;
5,3,1;
12,6,3;
Row n has 1+floor(n/2) terms.
T(5,2)=3 counts H(UD)(UD), (UD)H(UD) and (UD)(UD)H, where U=(1,1), H=(1,0), D=(1,-1) (the peaks at odd height are shown between parentheses).
CROSSREFS
Cf. A001006.
Sequence in context: A284464 A038041 A197591 * A097611 A135376 A132850
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Sep 03 2004
STATUS
approved