A166287 Number of peak plateaux in all Dyck paths of semilength n with no UUU's and no DDD's (U=(1,1), D=(1,-1)).

0, 0, 0, 1, 3, 8, 21, 53, 133, 334, 839, 2112, 5329, 13475, 34143, 86674, 220400, 561309, 1431522, 3655480, 9345287, 23916622, 61267207, 157088278, 403103955, 1035192885, 2660312103, 6841157380, 17603254230, 45321606641, 116748360064

Offset: 0

Author

Emeric Deutsch, Oct 12 2009

Keywords

• nonn

Comments

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

Examples

			a(4)=3 because we have UDUDUDUD, UDUDUUDD, UDUUDDUD, UD(UUDUDD), UUDDUDUD, UUDDUUDD, (UUDUDD)UD, (UUDUDUDD) (the 3 peak plateaux are shown between parentheses).
		

Links

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

Crossrefs

Cf. A166285.

Programs

Maple

h := sqrt((1-3*z+z^2)*(1+z+z^2)): G := ((1-z-z^2-h)*1/2)/((1-z)*h): Gser := series(G, z = 0, 35): seq(coeff(Gser, z, n), n = 0 .. 32);

Mathematica

CoefficientList[Series[((1-x-x^2-Sqrt[(1-3*x+x^2)*(1+x+x^2)])*1/2)/((1-x)*Sqrt[(1-3*x+x^2)*(1+x+x^2)]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)

PARI

x='x+O('x^50); concat([0,0,0], Vec(((1-x-x^2-sqrt((1-3*x+x^2)*(1+x+x^2))))/(2*(1-x)*sqrt((1-3*x+x^2)*(1+x+x^2))))) \\ G. C. Greubel, Mar 22 2017

Formula

a(n) = Sum_{k>=0} k*A166285(n,k).

G.f.: G=(1-z-z^2-h)/[2(1-z)h], where h = sqrt((1-3z+z^2)(1+z+z^2)).

a(n) ~ (3+sqrt(5))^n / (5^(1/4) * sqrt(Pi*n) * 2^(n+1)). - Vaclav Kotesovec, Mar 20 2014

Conjecture: n*a(n) +(-4*n+3)*a(n-1) +3*(n-1)*a(n-2) +(n-9)*a(n-3) +(3*n-5)*a(n-4) +(-3*n+7)*a(n-5) +(-2*n+13)*a(n-6) +(n-6)*a(n-7)=0. - R. J. Mathar, Jun 14 2016

Conjecture: n*(2*n-5)*(2*n-7)*a(n) -(2*n-7)*(6*n^2-17*n+8)*a(n-1) +(n-2)*(4*n^2-16*n-1)*a(n-2) +(-4*n^3+32*n^2-71*n+44)*a(n-3) +(2*n-3) (6*n^2-37*n+54)*a(n-4) -(n-4)*(2*n-3)*(2*n-5)*a(n-5)=0. - R. J. Mathar, Jun 14 2016