|
| |
|
|
A119012
|
|
Number of valleys strictly above the x-axis in all hill-free Dyck paths of semilength n (a hill in a Dyck path is a peak at level 1).
|
|
1
| |
|
|
0, 0, 1, 5, 23, 98, 405, 1644, 6604, 26356, 104746, 415155, 1642493, 6490622, 25629581, 101156936, 399151400, 1574818496, 6213255614, 24515233082, 96739530062, 381803092580, 1507141137026, 5950525214360, 23498966702808
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,4
|
|
|
COMMENTS
| a(n)=Sum(k*A119011(n,k),k=0..n-2).
|
|
|
REFERENCES
| E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
|
|
|
FORMULA
| G.f.=2[1-3z-(1-z)sqrt(1-4z)]/[(1+2z+sqrt(1-4z))^2*sqrt(1-4z)].
|
|
|
EXAMPLE
| a(4)=5 because in the 6 (=A000957(5)) hill-free Dyck paths of semilength 4, namely UUUUDDDD, UUUD|UDDD, UUD|UUDDD, UUD|UD|UDD, UUUDD|UDD and UUDDUUDD (U=(1,1), D=(1,-1)) we have altogether 5 valleys strictly above the x-axis (indicated by |).
|
|
|
MAPLE
| G:=2*(1-3*z-(1-z)*sqrt(1-4*z))/(1+2*z+sqrt(1-4*z))^2/sqrt(1-4*z): Gser:=series(G, z=0, 35): seq(coeff(Gser, z, n), n=1..28);
|
|
|
CROSSREFS
| Cf. A119011.
Sequence in context: A140529 A055489 A109765 * A084615 A181331 A196489
Adjacent sequences: A119009 A119010 A119011 * A119013 A119014 A119015
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), May 08 2006
|
| |
|
|
