A097892 Triangle read by rows: T(n,k) is the number of Motzkin paths of length n and having k peaks at even height.

1, 1, 2, 4, 8, 1, 17, 4, 38, 12, 1, 88, 34, 5, 209, 95, 18, 1, 506, 264, 59, 6, 1244, 731, 187, 25, 1, 3097, 2020, 582, 92, 7, 7791, 5578, 1786, 322, 33, 1, 19773, 15404, 5420, 1096, 134, 8, 50563, 42558, 16308, 3652, 510, 42, 1, 130149, 117652, 48744, 11960, 1872

Offset

0,3

Comments

Row sums are the Motzkin numbers (A001006).

Links

Table of n, a(n) for n=0..55.

Formula

G.f.=G=G(t, z) satisfies z^2*(1-z)G^2-(1-z)(1-z+z^2-tz^2)G+1-z+z^2-tz^2=0.

Example

Triangle begins:
1;
1;
2;
4;
8,1;
17,4;
38,12,1;
Row n (n>=2) contains floor(n/2) terms.
T(5,1)=4 counts HU(UD)D, U(UD)DH, UH(UD)D and U(UD)HD, where U=(1,1), H=(1,0), D=(1,-1) (the peaks at even height are shown between parentheses).

Crossrefs

Cf. A001006.

Sequence in context: A030275 A097874 A097885 * A197282 A215452 A362985

Adjacent sequences: A097889 A097890 A097891 * A097893 A097894 A097895

Keyword

nonn,tabf

Author

Emeric Deutsch, Sep 03 2004

Status

approved