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

Author

Emeric Deutsch, Oct 10 2008

Keywords

• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A143952, A143953, A079309.

Programs

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 *)

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

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