A121525 Number of up steps starting 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, 5, 19, 67, 219, 690, 2110, 6322, 18639, 54268, 156398, 446960, 1268351, 3577679, 10039583, 28046201, 78039545, 216388938, 598136340, 1648730940, 4533180211, 12435470410, 34042090044, 93012717072, 253692955789

Offset

1,3

Comments

a(n)=Sum(k*A121524(n,k), k=0..n-1). a(n)+A121523(n)=n*fibonacci(2n-1).

Links

Table of n, a(n) for n=1..26.

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

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

Example

a(3)=5 because we have UDUDUD, UDU(U)DD, U(U)DDUD, U(U)D(U)DD and U(U)UDDD, the up steps starting at an odd level being shown between parentheses (U=(1,1), D=(1,-1)).

Maple

G:=z^2*(1-z-2*z^2+3*z^3-2*z^4)/(1+z)/(1-3*z+z^2)^2/(1-z-z^2): Gser:=series(G, z=0, 34): seq(coeff(Gser, z, n), n=1..30);

Mathematica

Rest[CoefficientList[Series[x^2*(1-x-2*x^2+3*x^3-2*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. A121523, A121524, A001519.

Sequence in context: A067325 A273599 A347311 * A163872 A035344 A114277

Adjacent sequences: A121522 A121523 A121524 * A121526 A121527 A121528

Keyword

nonn

Author

Emeric Deutsch, Aug 05 2006

Status

approved