A167639 Number of peaks at even level in all Dyck paths of semilength n that have no ascents and no descents of length 1.

0, 0, 1, 0, 3, 2, 12, 16, 59, 110, 325, 716, 1926, 4584, 11887, 29328, 75071, 188462, 480778, 1217876, 3107689, 7913082, 20221903, 51664040, 132259190, 338721180, 868587021, 2228677360, 5723740309, 14709001454, 37826827606, 97335031824

Offset: 0

Author

Emeric Deutsch, Nov 08 2009

Keywords

• nonn

Comments

a(n) = Sum_{k>=0} k*A167637(n,k).

Examples

			a(5)=2 because U(UD)DUUUDDD, UUUDDDU(UD)D, UUUDDUUDDD, and UUUUUDDDDD have 1 + 1 + 0 + 0 = 2 even-level peaks (shown between parentheses).
		

Links

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

Crossrefs

Cf. A167636, A167637.

Programs

Maple

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

Mathematica

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

PARI

x='x+O('x^50); concat([0,0], Vec(x^2/((1+x-x^2)*sqrt((1+x+x^2)*(1-3*x+x^2))))) \\ G. C. Greubel, Feb 12 2017

Formula

G.f.: z^2/((1+z-z^2)*sqrt((1+z+z^2)*(1-3*z+z^2))).

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

Equivalently, a(n) ~ phi^(2*n - 1) / (4 * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 08 2021

D-finite with recurrence (-n+2)*a(n) +(n-3)*a(n-1) +2*(2*n-5)*a(n-2) +(n-5)*a(n-3) +(-3*n+11)*a(n-5) +(n-4)*a(n-6)=0. - R. J. Mathar, Jul 26 2022