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
Keywords
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
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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);
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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