A143954 Number of peaks in the peak plateaux of all Dyck paths of semilength n.

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

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: A359293 A092492 A070857 * A047145 A240525 A264200

Adjacent sequences: A143951 A143952 A143953 * A143955 A143956 A143957

Keyword

nonn

Author

Emeric Deutsch, Oct 10 2008

Status

approved