A191307 Sum of the heights of the first peaks in all dispersed Dyck paths of length n (i.e., in Motzkin paths of length n with no (1,0)-steps at positive heights).

0, 0, 1, 2, 6, 11, 26, 47, 103, 187, 397, 727, 1519, 2806, 5809, 10814, 22254, 41702, 85460, 161042, 329002, 622932, 1269578, 2413644, 4909788, 9367188, 19024888, 36408748, 73850908, 141714823, 287137498, 552320023, 1118042743, 2155201063, 4359162493, 8419091443

Offset

0,4

Links

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

Formula

a(n) = Sum_{k>=0} k*A191306(n,k).

G.f.: ((1-z-z^2)*sqrt(1-4*z^2) - (1-2*z)*(1+z-z^2))/(2*z^3*(1-z)*(1-2*z)).

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

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

Example

a(4)=6 because, denoting U=(1,1), D=(1,-1), H=(1,0), in HHHH, HHUD, HUDH, UDHH, UDUD, and UUDD the sum of the heights of the first peaks is 0+1+1+1+1+2=6.

Maple

g:=(((1-z-z^2)*sqrt(1-4*z^2)-(1-2*z)*(1+z-z^2))*1/2)/(z^3*(1-z)*(1-2*z)): gser:=series(g, z=0, 40): seq(coeff(gser, z, n), n=0..35);

Mathematica

CoefficientList[Series[(((1-x-x^2)*Sqrt[1-4*x^2]-(1-2*x)*(1+x-x^2))*1/2) /(x^3*(1-x)*(1-2*x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)

Prog

(PARI) x='x+O('x^50); concat([0, 0], Vec(((1-z-z^2)*sqrt(1-4*z^2) - (1-2*z)*(1+z-z^2))/(2*z^3*(1-z)*(1-2*z)))) \\ G. C. Greubel, Mar 26 2017

Crossrefs

Cf. A191306.

Sequence in context: A079118 A211054 A034466 * A007186 A033304 A091622

Adjacent sequences: A191304 A191305 A191306 * A191308 A191309 A191310

Keyword

nonn

Author

Emeric Deutsch, May 30 2011

Status

approved