|
| |
|
|
A135337
|
|
Number of Dyck paths of semilength n with no DUUU's starting at level 1.
|
|
1
| |
|
|
1, 1, 2, 5, 13, 36, 105, 320, 1011, 3289, 10957, 37216, 128435, 449142, 1588228, 5669505, 20403322, 73945553, 269647630, 988642372, 3642310793, 13476857235, 50059454347, 186598634398, 697777187275, 2616919372356, 9840647362248
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
COMMENTS
| Column 0 of A135331. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 14 2007
|
|
|
REFERENCES
| A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.
|
|
|
FORMULA
| G.f.=1+zC^2/(1+z^3*C^4)=(1-z)(2C-1)/[(1-2z)C+z], where C=[1-sqrt(1-4z)]/(2z) is the g.f. of the Catalan numbers (A000108). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 14 2007
From Gary W. Adamson, Jul 26 2011: (start) a(n) = sum of top row terms of M^(n-1), M = an infinite square production matrix as follows, in which a column of [1,1,0,0,0,...] is prepended to an infinite lower triangular matrix of all 1's and the rest zeros:
1, 1, 0, 0, 0, 0,...
1, 1, 1, 0, 0, 0,...
0, 1, 1, 1, 0, 0,...
0, 1, 1, 1, 1, 0,...
0, 1, 1, 1, 1, 1,...
... (end)
|
|
|
EXAMPLE
| a(4)=13 because among the 14 (=A000108(4)) Dyck paths of semilength 14 only UDUUUDDD does not qualify.
a(4) = 13 since the top row of M^3 = [4, 5, 3, 1, 0, 0, 0,...] with 13 = (4 + 5 + 3 + 1).
|
|
|
MAPLE
| G:=(2*C-1)/(C-z*(2*C-1)): C:=((1-sqrt(1-4*z))*1/2)/z: Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=0..27); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 14 2007
|
|
|
CROSSREFS
| Cf. A000108, A135331.
Sequence in context: A125094 A114465 A135310 * A133365 A135335 A066723
Adjacent sequences: A135334 A135335 A135336 * A135338 A135339 A135340
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Dec 07 2007
|
|
|
EXTENSIONS
| More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 14 2007
|
| |
|
|
