|
| |
|
|
A128096
|
|
Number of steps that touch the x-axis in all peakless Motzkin paths of length n.
|
|
1
| |
|
|
1, 2, 5, 12, 27, 62, 144, 336, 790, 1870, 4452, 10656, 25629, 61910, 150145, 365450, 892434, 2185928, 5369097, 13221422, 32634935, 80730942, 200116410, 496992992, 1236482727, 3081389406, 7690966549, 19224282880, 48119034729, 120599916654
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,2
|
|
|
COMMENTS
| a(n)=Sum(k*A128095(n,k), k=1..n).
|
|
|
FORMULA
| G.f.=4[1-z^2-sqrt((1+z+z^2)(1-3z+z^2))]/[1-z+z^2+sqrt((1+z+z^2)(1-3z+z^2))]^2.
|
|
|
EXAMPLE
| a(3)=5 because in the peakless Motzkin paths of length 3 (namely HHH and UHD, where H=(1,0), U=(1,1) and D=(1,-1)) all the steps, with the exception of H in UHD, touch the x-axis.
|
|
|
MAPLE
| g:=4*(1-z^2-sqrt((1+z+z^2)*(1-3*z+z^2)))/(1-z+z^2+sqrt((1+z+z^2)*(1-3*z+z^2)))^2: gser:=series(g, z=0, 38): seq(coeff(gser, z, n), n=1..35);
|
|
|
CROSSREFS
| Cf. A128095.
Sequence in context: A091596 A077863 A018009 * A018010 A026710 A171579
Adjacent sequences: A128093 A128094 A128095 * A128097 A128098 A128099
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 14 2007
|
| |
|
|
