A097887 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.).

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

Offset

0,6

Comments

Row sums are the Motzkin numbers (A001006). Column 0 gives A089380.

Links

Table of n, a(n) for n=0..59.

Formula

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.

Example

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).

Crossrefs

Cf. A001006, A089380.

Sequence in context: A368826 A352150 A161826 * A019761 A188614 A290798

Adjacent sequences: A097884 A097885 A097886 * A097888 A097889 A097890

Keyword

nonn,tabf

Author

Emeric Deutsch, Sep 02 2004

Status

approved