The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
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!)
A121523 Number of up steps starting 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
1, 3, 10, 33, 103, 315, 941, 2770, 8051, 23171, 66138, 187486, 528365, 1481501, 4135756, 11500721, 31871625, 88054825, 242609585, 666783380, 1828452021, 5003697403, 13667302500, 37267071708, 101455834153, 275797332135 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
a(n) = Sum(k*A121522(n,k), k=1..n). a(n)+A121525(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(1-3z+z^2+5z^3-5z^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)=10 because we have (U)D(U)D(U)D, (U)D(U)UDD, (U)UDD(U)D, (U)UDUDD and (U)U(U)DDD, the up steps starting at even level being shown between parentheses (U=(1,1), D=(1,-1)).
MAPLE
G:=z*(1-3*z+z^2+5*z^3-5*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*(1-3*x+x^2+5*x^3-5*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: A316411 A292549 A062454 * A115240 A027989 A096483
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 | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 29 03:48 EDT 2024. Contains 372921 sequences. (Running on oeis4.)