|
Search: id:A129164
|
|
|
| A129164 |
|
Sum of pyramid weights in all skew Dyck paths of semilength n. A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps. A pyramid in a skew Dyck word (path) is a factor of the form U^h D^h, h being the height of the pyramid. A pyramid in a skew Dyck word w is maximal if, as a factor in w, it is not immediately preceded by a U and immediately followed by a D. The pyramid weight of a skew Dyck path (word) is the sum of the heights of its maximal pyramids. |
|
+0
2
|
|
| 1, 5, 22, 97, 436, 1994, 9241, 43257, 204052, 968440, 4619011, 22120630, 106300507, 512321437, 2475395302, 11986728457, 58156146652, 282640193312, 1375737276787, 6705522150972, 32724071280517, 159878425878847
(list; graph; listen)
|
|
|
OFFSET
|
1,2
|
|
|
COMMENT
|
a(n)=Sum(k*A129163(n,k),k=1..n). Partial sums of A026378.
|
|
REFERENCES
|
E. Deutsch, E. Munarini and S. Rinaldi, Skew Dyck paths (in preparation).
A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137, 1995, 155-176).
|
|
FORMULA
|
G.f.=[1/sqrt(1-6z+5z^2)-1/(1-z)]/2.
|
|
EXAMPLE
|
a(2)=5 because the pyramid weights of the paths (UD)(UD), (UUDD) and U(UD)L are 2, 2 and 1, respectively (the maximal pyramids are shown between parentheses).
|
|
MAPLE
|
G:=(1/sqrt(1-6*z+5*z^2)-1/(1-z))/2: Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=1..26);
|
|
CROSSREFS
|
Cf. A129163, A026378.
Adjacent sequences: A129161 A129162 A129163 this_sequence A129165 A129166 A129167
Sequence in context: A026888 A083586 A129158 this_sequence A123347 A087439 A033452
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 03 2007
|
|
|
Search completed in 0.002 seconds
|
