|
Search: id:A110190
|
|
|
| A110190 |
|
Number of (1,0)-steps on the lines y=0 and y=1 in all Schroeder paths of length 2n (a Schroeder path of length 2n is a path from (0,0) to (2n,0), consisting of steps U=(1,1), D=(1,-1) and H=(2,0) and never going below the x-axis). |
|
+0
2
|
|
| 0, 1, 5, 24, 116, 568, 2820, 14184, 72180, 371112, 1925380, 10068728, 53023860, 280969560, 1497072132, 8016213960, 43114424308, 232817773640, 1261793848836, 6861179441880, 37421756333172, 204671007577464, 1122275850740996
(list; graph; listen)
|
|
|
OFFSET
|
0,3
|
|
|
COMMENT
|
a(n)=sum(k*A110189(n,k), k=0..n).
|
|
FORMULA
|
G.f.=z(1-z-2zR+z^2+2z^2*R+z^2*R^2)/(1-3z-zR+z^2+z^2*R)^2, where R=1+zR+zR^2={1-z-sqrt(1-6z+z^2)]/(2z) is the g.f. for the large Schroeder numbers (A006318).
|
|
EXAMPLE
|
a(2)=5 because in the 6 (=A006318(2)) Schroeder paths of length 4, namely, HH, HUD, UDH, UDUD, UHD, UUDD, all 5 H-steps are at levels 0 or 1.
|
|
MAPLE
|
R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=z*(1-z-2*z*R+z^2+2*z^2*R+z^2*R^2)/(1-3*z-z*R+z^2+z^2*R)^2: Gser:=series(G, z=0, 30): 0, seq(coeff(Gser, z^n), n=1..26);
|
|
CROSSREFS
|
Cf. A006318, A110189.
Sequence in context: A004254 A086347 A026707 this_sequence A026784 A017977 A017978
Adjacent sequences: A110187 A110188 A110189 this_sequence A110191 A110192 A110193
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 15 2005
|
|
|
Search completed in 0.002 seconds
|
