A191309 Number of peaks at height >= 2 in all dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0) steps at positive heights).

0, 0, 0, 0, 1, 2, 8, 16, 47, 94, 244, 488, 1186, 2372, 5536, 11072, 25147, 50294, 112028, 224056, 491870, 983740, 2135440, 4270880, 9188406, 18376812, 39249768, 78499536, 166656772, 333313544, 704069248, 1408138496, 2961699667, 5923399334, 12412521388, 24825042776

Offset

0,6

Comments

Also number of valleys (i.e., DU's) in all dispersed Dyck paths of length n. Example: a(4)=1 because in HHHH, HHUD, HUDH, UDHH, UDUD, and UUDD we have 0+0+0+0+1+0 = 1 valley.

Also number of doublerises (i.e., UU's) in all dispersed Dyck paths of length n. Example: a(4)=1 because in HHHH, HHUD, HUDH, UDHH, UDUD, and UUDD we have 0+0+0+0+0+1 = 1 doublerise.

Links

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

Formula

a(2*n+1) = 2*a(2*n).

a(2*n+4) = A029760(n).

G.f.: g = 2*z^2*(1-q)/(q*(1-2*z+q)^2), where q=sqrt(1-4*z^2).

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

Conjecture: n*(n-4)*a(n) +(n^2-10*n+15)*a(n-1) +2*(-5*n^2+28*n-27)*a(n-2) -4*(n-3)*(n-8) *a(n-3) +24*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Jun 14 2016

Example

a(4)=1 because in HHHH, HHUD, HUDH, UDHH, UDUD, and UUDD we have 0+0+0+0+0+1 =1 peak at height >=2.

Maple

q := sqrt(1-4*z^2): g := 2*z^2*(1-q)/(q*(1-2*z+q)^2): gser := series(g, z = 0, 40): seq(coeff(gser, z, n), n = 0 .. 35);

Mathematica

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

Prog

(PARI) x='x+O('x^50); concat([0, 0, 0, 0], Vec(2*x^2*(1-sqrt(1-4*x^2))/(sqrt(1-4*x^2)*(1-2*x+ sqrt(1-4*x^2))^2))) \\ G. C. Greubel, Mar 26 2017

Crossrefs

Cf. A029760, A191308.

Sequence in context: A176143 A296946 A096227 * A323351 A134353 A280229

Adjacent sequences: A191306 A191307 A191308 * A191310 A191311 A191312

Keyword

nonn

Author

Emeric Deutsch, May 30 2011

Status

approved