A121530 Number of double rises at an 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.

0, 1, 4, 14, 47, 148, 454, 1359, 4004, 11644, 33521, 95696, 271300, 764605, 2143964, 5985186, 16643779, 46124692, 127433562, 351106955, 964976460, 2646158176, 7241414949, 19779499584, 53933402472, 146828245753, 399137621524

Offset

1,3

Comments

a(n)=Sum(k*A121529(n,k), k>=0). a(n)+A121532(n)=A054444(n-2).

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^2*(1-2z-z^2+4z^3-3z^4)/[(1+z)(1-3z+z^2)^2*(1-z-z^2)].

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

Equivalently, a(n) ~ phi^(2*n-2) * n / 5, where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 06 2021

Example

a(3)=4 because we have UDUDUD, UDU/UDD, U/UDDUD, U/UDUDD and U/UUDDD, the double rises at an odd level being indicated by a / (U=(1,1), D=(1,-1)).

Maple

g:=z^2*(1-2*z-z^2+4*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^2*(1-2*x-x^2+4*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. A121529, A121532, A054444.

Sequence in context: A258255 A124805 A362279 * A121299 A326346 A046718

Adjacent sequences: A121527 A121528 A121529 * A121531 A121532 A121533

Keyword

nonn

Author

Emeric Deutsch, Aug 05 2006

Status

approved