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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. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
a(n)=Sum(k*A121524(n,k), k=0..n-1). a(n)+A121523(n)=n*fibonacci(2n-1).
LINKS
E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
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
Sequence in context: A067325 A273599 A347311 * A163872 A372884 A035344
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Aug 05 2006
STATUS
approved

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Last modified July 14 15:30 EDT 2024. Contains 374322 sequences. (Running on oeis4.)