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A143954 Number of peaks in the peak plateaux of all Dyck paths of semilength n. 2
0, 0, 1, 5, 19, 68, 243, 880, 3233, 12021, 45119, 170595, 648787, 2479057, 9509627, 36598497, 141246127, 546433952, 2118424887, 8227983472, 32010173957, 124715628852, 486550020967, 1900433894942, 7431033132717, 29085434212042 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

FORMULA

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

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

a(n) ~ 2^(2*n+1)/(9*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 20 2014

a(n) = Sum_{k=1..n-1} A079309(k). - Doug Bell, Jun 23 2015

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

EXAMPLE

a(3)=5 because in the peak plateaux of the Dyck paths UDUDUD, UD(UUDD), (UUDD)UD, (UUDUDD) and U(UUDD)D, shown between parentheses, we have 0 + 1 + 1 + 2 + 1 = 5 peaks.

MAPLE

C:=((1-sqrt(1-4*z))*1/2)/z: G:=z^2*C/((1-z)^2*sqrt(1-4*z)): Gser:=series(G, z= 0, 30): seq(coeff(Gser, z, n), n=0..25);

MATHEMATICA

CoefficientList[Series[x^2*((1-Sqrt[1-4*x])*1/2)/x/((1-x)^2*Sqrt[1-4*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)

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

CROSSREFS

Cf. A143952, A143953, A079309.

Sequence in context: A001435 A092492 A070857 * A047145 A240525 A264200

Adjacent sequences:  A143951 A143952 A143953 * A143955 A143956 A143957

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Oct 10 2008

STATUS

approved

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Last modified April 5 02:41 EDT 2020. Contains 333238 sequences. (Running on oeis4.)