|
| |
|
|
A114583
|
|
Triangle read by rows: T(n,k) is the number of Motzkin paths of length n and having k UHD's, where U=(1,1),H=(1,0),D=(1,-1) (0<=k<=floor(n/3)).
|
|
1
| |
|
|
1, 1, 2, 3, 1, 7, 2, 15, 6, 36, 14, 1, 85, 39, 3, 209, 102, 12, 517, 280, 37, 1, 1303, 758, 123, 4, 3312, 2085, 381, 20, 8510, 5730, 1194, 76, 1, 22029, 15849, 3657, 295, 5, 57447, 43914, 11187, 1056, 30, 150709, 122090, 33903, 3734, 135, 1, 397569, 340104
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
COMMENTS
| Row n contains 1+floor(n/3) terms. Row sums are the Motzkin numbers (A001006). Column 1 yields A114584. Sum(k*T(n,k),k=0..floor(n/3))=A005717(n-2).
|
|
|
FORMULA
| G.f.=G=G(t, z) satisfies G=1+zG+z^2*G(tz-z+G).
|
|
|
EXAMPLE
| T(5,1)=6 because we have HH(UHD), UD(UHD), (UHD)HH, (UHD)UD, H(UHD)H and U(UHD)D, where U=(1,1),H=(1,0),D=(1,-1) (the UHD's are shown between parentheses).
Triangle begins:
1;
1;
2;
3,1;
7,2;
15,6;
36,14,1;
|
|
|
MAPLE
| G:=(1-z-t*z^3+z^3-sqrt((1-3*z+z^3-t*z^3)*(1+z+z^3-t*z^3)))/2/z^2: Gser:=simplify(series(G, z=0, 20)): P[0]:=1: for n from 1 to 17 do P[n]:=sort(coeff(Gser, z^n)) od: for n from 0 to 17 do seq(coeff(t*P[n], t^j), j=1..1+floor(n/3)) od; # yields sequence in triangular form
|
|
|
CROSSREFS
| Cf. A001006, A114584, A115717.
Sequence in context: A128264 A114858 A193491 * A114581 A085588 A118008
Adjacent sequences: A114580 A114581 A114582 * A114584 A114585 A114586
|
|
|
KEYWORD
| nonn,tabf
|
|
|
AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 09 2005
|
| |
|
|
