|
Search: id:A110110
|
|
|
| A110110 |
|
Number of symmetric Schroeder paths of length 2n (A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis). |
|
+0
2
|
|
| 1, 2, 4, 8, 18, 38, 88, 192, 450, 1002, 2364, 5336, 12642, 28814, 68464, 157184, 374274, 864146, 2060980, 4780008, 11414898, 26572086, 63521352, 148321344, 354870594, 830764794, 1989102444, 4666890936, 11180805570, 26283115038
(list; graph; listen)
|
|
|
OFFSET
|
0,2
|
|
|
COMMENT
|
a(n)=A026003(n-1)+A026003(n) (n>=1; indeed, every symmetric Schroeder path of length 2n is either a left factor L of length n-1 of a Schroeder path, followed by a H=(2,0) step and followed by the mirror image of L, or it is a left factor of length n of a Schroeder paths, followed by its mirror image).
|
|
FORMULA
|
G.f.=(1+z)R(z^2)/[1-zR(z^2)], where R=1+zR+zR^2=[1-z-sqrt(1-6z+z^2)]/(2z) is the g.f. of the large Schroeder numbers.
|
|
EXAMPLE
|
a(2)=4 because we have HH, UDUD, UHD and UUDD.
|
|
MAPLE
|
RR:=(1-z^2-sqrt(1-6*z^2+z^4))/2/z^2: G:=(1+z)*RR/(1-z*RR): Gser:=series(G, z=0, 36): 1, seq(coeff(Gser, z^n), n=1..33);
|
|
CROSSREFS
|
Cf. A026003, A006318.
Sequence in context: A092507 A024415 A018096 this_sequence A056362 A086585 A052910
Adjacent sequences: A110107 A110108 A110109 this_sequence A110111 A110112 A110113
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 12 2005
|
|
|
Search completed in 0.002 seconds
|
