login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A121532 Number of double rises at an even 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
0, 0, 1, 6, 24, 87, 290, 926, 2861, 8640, 25634, 75015, 217100, 622620, 1772097, 5011394, 14093980, 39448623, 109954398, 305344314, 845165725, 2332485420, 6420202246, 17629525871, 48304680504, 132092031672, 360557665825 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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

a(n) = Sum_{k>=0} k*A121531(n,k).

a(n) = A054444(n-2) - A121530(n).

G.f.: x^3*(1-3*x^2+2*x^3-x^4)/((1+x)*(1-3*x+x^2)^2*(1-x-x^2)). [Corrected by Georg Fischer, May 24 2019]

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

EXAMPLE

a(3)=1 because we have UDUDUD, UDUUDD, UUDDUD, UUDUDD and UU/UDDD, the double rises at an odd level being indicated by a / (U=(1,1), D=(1,-1)).

MAPLE

g:=z^3*(1-3*z^2+2*z^3-z^4)/(1+z)/(1-3*z+z^2)^2/(1-z-z^2): gser:=series(g, z=0, 35): seq(coeff(gser, z, n), n=1..32);

MATHEMATICA

Rest[CoefficientList[Series[x^3*(1-3*x^2+2*x^3-x^4)/(1+x)/(1-3*x+x^2)^2/(1-x-x^2), {x, 0, 30}], x]] (* Vaclav Kotesovec, Mar 20 2014 *)

PROG

(PARI) my(x='x+O('x^30)); concat([0, 0], Vec(x^3*(1-3*x^2+2*x^3-x^4)/((1+x)*(1-3*x+x^2)^2*(1-x-x^2)))) \\ G. C. Greubel, May 24 2019

(MAGMA) R<x>:=PowerSeriesRing(Integers(), 30); [0, 0] cat Coefficients(R!( x^3*(1-3*x^2+2*x^3-x^4)/((1+x)*(1-3*x+x^2)^2*(1-x-x^2)) )); // G. C. Greubel, May 24 2019

(Sage) a=(x^3*(1-3*x^2+2*x^3-x^4)/((1+x)*(1-3*x+x^2)^2*(1-x-x^2)) ).series(x, 30).coefficients(x, sparse=False); a[1:] # G. C. Greubel, May 24 2019

CROSSREFS

Cf. A121530, A121531, A054444.

Sequence in context: A166060 A124807 A271789 * A025472 A255474 A249976

Adjacent sequences:  A121529 A121530 A121531 * A121533 A121534 A121535

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Aug 05 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 | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 16 08:15 EDT 2019. Contains 328051 sequences. (Running on oeis4.)