|
Search: id:A118973
|
|
|
| A118973 |
|
Number of hill-free Dyck paths of semilength n+2 and having length of first descent equal to 2 (a hill in a Dyck path is a peak at level 1). |
|
+0
3
|
|
| 1, 0, 2, 5, 16, 51, 168, 565, 1934, 6716, 23604, 83806, 300154, 1083137, 3934404, 14374413, 52787766, 194746632, 721435884, 2682522918, 10008240456, 37455101382, 140569122624, 528926230530, 1994980278636, 7541234323096
(list; graph; listen)
|
|
|
OFFSET
|
0,3
|
|
|
COMMENT
|
Also, for a given j>=2, number of hill-free Dyck paths of semilength n+j and having length of first descent equal to j. a(n)=A000108(n+1)-A000108(n)-[A000957(n+2)-A000957(n+1)]. Columns 2,3,4,... of A118972 (without the initial 0's).
|
|
REFERENCES
|
E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241, 241-265, 2001.
|
|
FORMULA
|
G.f.=(1-z)CF, where F=[1-sqrt(1-4z)]/[z(3-sqrt(1-4z)] and C=[1-sqrt(1-4z)]/(2z) is the Catalan function.
|
|
EXAMPLE
|
a(2)=2 because we have uu(dd)uudd and uuu(dd)udd, where u=(1,1),d=(1,-1) (the first descents are shown between parentheses).
|
|
MAPLE
|
F:=(1-sqrt(1-4*z))/z/(3-sqrt(1-4*z)): C:=(1-sqrt(1-4*z))/2/z: g:=(1-z)*C*F: gser:=series(g, z=0, 33): seq(coeff(gser, z, n), n=0..28);
|
|
CROSSREFS
|
Cf. A000108, A000957, A118972, A001558.
Sequence in context: A148383 A148384 A148385 this_sequence A148386 A148387 A121651
Adjacent sequences: A118970 A118971 A118972 this_sequence A118974 A118975 A118976
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Emeric Deutsch (deutsch(AT)duke.poly.edu), May 08 2006
|
|
|
Search completed in 0.002 seconds
|
