login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A121483 Number of peaks at odd level in all nondecreasing Dyck paths of semilength n. A nondecreasing Dyck path is a Dyck path for which the sequence of the altitudes of the valleys is nondecreasing. 2
1, 2, 6, 19, 56, 167, 487, 1411, 4047, 11527, 32617, 91790, 257065, 716896, 1991792, 5515535, 15227846, 41930133, 115176023, 315676425, 863475561, 2357539227, 6425887551, 17487572124, 47522431681, 128969086382, 349567320762 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) = Sum(k*A121481(n,k),k=0..n).

LINKS

Table of n, a(n) for n=1..27.

E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.

Index entries for linear recurrences with constant coefficients, signature (6,-9,-5,15,-1,-4,1).

FORMULA

G.f.: z(1-z)(1-3z+6z^3-3z^4)/[(1+z)(1-3z+z^2)^2*(1-z-z^2)].

Recurrence: (n^2 - 5*n - 20)*a(n) = (3*n^2 - 12*n - 79)*a(n-1) + (n^2 - 7*n - 16)*a(n-2) - (5*n^2 - 19*n - 138)*a(n-3) - (n^2 - 6*n - 31)*a(n-4) + (n^2 - 3*n - 24)*a(n-5). - Vaclav Kotesovec, Mar 20 2014

a(n) ~ (sqrt(5)-1) * (3+sqrt(5))^n * n / (5*2^(n+2)). - Vaclav Kotesovec, Mar 20 2014

EXAMPLE

a(2)=2 because in UDUD and UUDD we have altogether 2 peaks at odd level; here U=(1,1) and D=(1,-1).

MAPLE

G:=z*(1-z)*(1-3*z+6*z^3-3*z^4)/(1+z)/(1-3*z+z^2)^2/(1-z-z^2): Gser:=series(G, z=0, 33): seq(coeff(Gser, z, n), n=1..30);

MATHEMATICA

Rest[CoefficientList[Series[x*(1-x)*(1-3*x+6*x^3-3*x^4)/(1+x)/(1-3*x+x^2)^2/(1-x-x^2), {x, 0, 20}], x]] (* Vaclav Kotesovec, Mar 20 2014 *)

CROSSREFS

Cf. A121481, A121486, A038731.

Sequence in context: A183305 A192715 A226433 * A077834 A325918 A307564

Adjacent sequences: A121480 A121481 A121482 * A121484 A121485 A121486

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Aug 02 2006

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 7 09:43 EST 2023. Contains 360115 sequences. (Running on oeis4.)