The On-Line Encyclopedia of Integer Sequences (OEIS)

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A143954

Number of peaks in the peak plateaux of all Dyck paths of semilength n.
(history; published version)

#24 by Bruno Berselli at Thu Mar 23 04:36:49 EDT 2017

STATUS

proposed

approved

#23 by Michel Marcus at Thu Mar 23 01:04:19 EDT 2017

STATUS

editing

proposed

#22 by Michel Marcus at Thu Mar 23 01:04:16 EDT 2017

NAME

Number of peaks in the peak plateaux of all Dyck paths of semilength n. A peak plateau is a run of consecutive peaks that is preceded by an upstep and followed by a down step; a peak consists of an upstep followed by a downstep.

Number of peaks in the peak plateaux of all Dyck paths of semilength n.

COMMENTS

A peak plateau is a run of consecutive peaks that is preceded by an upstep and followed by a down step; a peak consists of an upstep followed by a downstep.

STATUS

proposed

editing

#21 by G. C. Greubel at Wed Mar 22 23:07:22 EDT 2017

STATUS

editing

proposed

#20 by G. C. Greubel at Wed Mar 22 23:07:12 EDT 2017

COMMENTS

a(n) = Sum(k*A143953(n,k), k=0..n-1).

LINKS

G. C. Greubel, <a href="/A143954/b143954.txt">Table of n, a(n) for n = 0..1000</a>

FORMULA

a(n) = Sum_{k=0..n-1} k*A143953(n,k).

PROG

(PARI) x='x+O('x^50); concat([0, 0], Vec(x*(1-sqrt(1-4*x))/(2*(1-x)^2*sqrt(1-4*x)))) \\ G. C. Greubel, Mar 22 2017

STATUS

approved

editing

#19 by R. J. Mathar at Thu Jun 16 11:14:53 EDT 2016

STATUS

editing

approved

#18 by R. J. Mathar at Thu Jun 16 11:14:45 EDT 2016

FORMULA

Conjecture: (-n+1)*a(n) +2*(3*n-4)*a(n-1) +(-9*n+13)*a(n-2) +2*(2*n-3)*a(n-3)=0. - _R. J. Mathar_, Jun 16 2016

. - R. J. Mathar, Jun 16 2016

#17 by R. J. Mathar at Thu Jun 16 11:14:24 EDT 2016

FORMULA

Conjecture: (-n+1)*a(n) +2*(3*n-4)*a(n-1) +(-9*n+13)*a(n-2) +2*(2*n-3)*a(n-3)=0

. - R. J. Mathar, Jun 16 2016

STATUS

approved

editing

#16 by Alois P. Heinz at Wed Jun 24 08:33:52 EDT 2015

STATUS

editing

approved

#15 by Alois P. Heinz at Wed Jun 24 08:31:54 EDT 2015

FORMULA

G.f. = : z^2*C/[(1-z)^2*sqrt(1-4z)], where C = [1-sqrt(1-4z)]/(2z) is the Catalan function.

For n > 0, a(n) = sum of first Sum_{k=1..n-1 terms of } A079309(k). - Doug Bell, Jun 23 2015

STATUS

proposed

editing

Discussion

Wed Jun 24

08:33

Alois P. Heinz: Changed your text into a formula. This is true for all n because the empty sum gives 0.