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A048285 Number of Dyck paths of length 2n with nondecreasing peaks. 9
1, 1, 2, 4, 9, 21, 51, 126, 316, 800, 2040, 5230, 13464, 34773, 90035, 233590, 607011, 1579438, 4114014, 10725109, 27979704, 73035818, 190737623, 498320800, 1302341411, 3404552915, 8902154847, 23281653957, 60897957049, 159312797657 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The name refers to weakly increasing peaks. The case of strictly increasing peaks is counted by A008930. - David Callan, Feb 18 2004

a(n) ~ 0.11997*[(3+sqrt(5))/2]^n (Theorem 2 of the Penaud-Roques paper). - Emeric Deutsch, Mar 05 2008

Row sums of A138155. - Emeric Deutsch, Mar 05 2008

For a constant 0.1199765127480778967304984... see A239528. - Vaclav Kotesovec, Mar 21 2014

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..690 (terms n=1..300 from Vaclav Kotesovec)

Sergi Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020.

J. G. Penaud and O. Roques, Génération de chemins de Dyck à pics croissants, Discrete Mathematics, Vol. 246, no. 1-3 (2002), 255-267.

FORMULA

G.f.: 1 + Sum_{n>=0} ((-1)^n x^{2n+1}(1-x)) / (Product_{i=1...n+1} ((1-x)(1-x^i)-x)).

Conjectural g.f.: Sum_{n>=0} (x*(1 - x))^n/( Product_{i=2..n+1} (1 - 2*x + x^i) ) (checked up to x^50). - Peter Bala, Mar 31 2017

EXAMPLE

a(3)=4 because we have UDUDUD, UDUUDD, UUDUDD and UUUDDD, where U=(1,1) and D=(1,-1).

MAPLE

g:= 1+sum((-1)^n*z^(2*n+1)*(1-z)/(product((1-z)*(1-z^i)-z, i=1..n+1)), n=0..40): gser:=series(g, z=0, 35): seq(coeff(gser, z, n), n=0..30); # Emeric Deutsch, Mar 05 2008

# second Maple program:

b:= proc(x, y, k, t) option remember; `if`(x=0, 1, `if`(y>0,

      `if`(t=1 and y>k, 0, b(x-1, y-1, `if`(t=1, min(k, y),

         k), 0)), 0) +`if`(y<x-1, b(x-1, y+1, k, 1), 0))

    end:

a:= n-> b(2*n, 0, n, 0):

seq(a(n), n=0..35);  # Alois P. Heinz, Jun 13 2017

# third Maple program:

b:= proc(n, i) option remember; `if`(n=0, 1, add(

      binomial(i, j)*add(b(n-2-(i-j)*2-2*t, i-j+t),

      t=0..n/2+j-i-1), j=0..i))

    end:

a:= n-> b(2*n, 0):

seq(a(n), n=0..35);  # Alois P. Heinz, Jun 13 2017

MATHEMATICA

Table[SeriesCoefficient[Sum[(-1)^k*x^(2*k+1)*(1-x)/Product[(1-x)*(1-x^i)-x, {i, 1, k+1}], {k, 0, n}], {x, 0, n}], {n, 1, 20}] (* Vaclav Kotesovec, Mar 21 2014 *)

CROSSREFS

Cf. A138155, A239528.

Sequence in context: A261232 A176334 A257386 * A051529 A230554 A005207

Adjacent sequences:  A048282 A048283 A048284 * A048286 A048287 A048288

KEYWORD

nonn,nice

AUTHOR

Olivier Roques (roques(AT)labri.u-bordeaux.fr)

EXTENSIONS

More terms from Emeric Deutsch, Mar 05 2008

a(0)=1 prepended by Alois P. Heinz, Jan 31 2017

STATUS

approved

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Last modified December 2 14:54 EST 2020. Contains 338877 sequences. (Running on oeis4.)