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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

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.

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

Cf. A001519, A121522, A121525.

Sequence in context: A316411 A292549 A062454 * A115240 A027989 A096483

Adjacent sequences:  A121520 A121521 A121522 * A121524 A121525 A121526

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Aug 05 2006

STATUS

approved

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Last modified May 29 07:24 EDT 2022. Contains 354122 sequences. (Running on oeis4.)