|
Search: id:A109191
|
|
|
| A109191 |
|
Triangle read by rows: T(n,k) is number of Grand Motzkin paths of length n having k hills (i.e. ud's starting at level 0). (A Grand Motzkin path is a path in the half-plane x>=0, starting at (0,0), ending at (n,0) and consisting of steps u=(1,1), d=(1,-1) and h=(1,0).). |
|
+0
2
|
|
| 1, 1, 2, 1, 5, 2, 13, 5, 1, 34, 14, 3, 91, 40, 9, 1, 247, 114, 28, 4, 678, 327, 87, 14, 1, 1877, 942, 267, 48, 5, 5233, 2723, 815, 161, 20, 1, 14674, 7892, 2478, 528, 75, 6, 41349, 22924, 7512, 1706, 270, 27, 1, 117001, 66712, 22718, 5452, 941, 110, 7, 332260
(list; graph; listen)
|
|
|
OFFSET
|
0,3
|
|
|
COMMENT
|
Row n contains 1+floor(n/2) terms. Row sums yield the central trinomial coefficients (A002426). T(n,0)=A109192(n). sum(k*T(n,k),k=0..floor(n/2))=A015518(n-1).
|
|
FORMULA
|
G.f. = 1/(1-z+z^2-tz^2-2z^2*M), where M=1+zM+z^2*M^2=[1-z-sqrt(1-2z-3z^2)]/(2z^2) is the g.f. of the Motzkin numbers (A001006).
|
|
EXAMPLE
|
T(3,1)=2 because we have hud and udh, where u=(1,1),d=(1,-1), h=(1,0).
Triangle begins:
1;
1;
2,1;
5,2;
13,5,1;
34,14,3;
91,40,9,1;
|
|
MAPLE
|
M:=(1-z-sqrt(1-2*z-3*z^2))/2/z^2: G:=1/(1-z+z^2-t*z^2-2*z^2*M): Gser:=simplify(series(G, z=0, 16)): P[0]:=1: for n from 1 to 14 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 14 do seq(coeff(t*P[n], t^k), k=1..1+floor(n/2)) od;
|
|
CROSSREFS
|
Cf. A002426, A109192, A015518, A001006.
Sequence in context: A082010 A113176 A113175 this_sequence A087123 A097131 A109738
Adjacent sequences: A109188 A109189 A109190 this_sequence A109192 A109193 A109194
|
|
KEYWORD
|
nonn,tabf
|
|
AUTHOR
|
Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 21 2005
|
|
|
Search completed in 0.002 seconds
|
