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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). 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified April 24 16:26 EDT 2019. Contains 322430 sequences. (Running on oeis4.)