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

Author

Emeric Deutsch, May 30 2011

Keywords

• nonn

Examples

			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.
		

Links

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

Crossrefs

Cf. A191306.

Programs

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 *)

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

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