A247287 Number of weak peaks in all Motzkin paths of length n. A weak peak of a Motzkin path is a vertex on the top of a hump. A hump is an upstep followed by 0 or more flatsteps followed by a downstep. For example, the Motzkin path u*duu*h*h*dd, where u=(1,1), h=(1,0), d(1,-1), has 4 weak peaks (shown by the stars).
0, 0, 1, 4, 13, 38, 108, 304, 857, 2426, 6902, 19728, 56622, 163092, 471205, 1365008, 3963321, 11530786, 33607190, 98105616, 286795300, 839470664, 2460038427, 7216652488, 21190820678, 62279238828, 183185851903, 539220930004, 1588341106957, 4681678922366
Offset: 0
Keywords
Examples
a(3)=4 because the Motzkin paths hhh, hu*d, u*dh, and u*h*d have 0, 1, 1, and 2 weak peaks (shown by the stars).
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
Programs
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Maple
g := ((1-z-sqrt(1-2*z-3*z^2))*(1/2))/((1-z)^2*sqrt(1-2*z-3*z^2)): gser := series(g, z = 0, 34): seq(coeff(gser, z, n), n = 0 .. 32);
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PARI
z='z+O('z^66); concat([0,0],Vec((1-z-sqrt(1-2*z-3*z^2))/(2*(1-z)^2*sqrt(1-2*z-3*z^2)))) \\ Joerg Arndt, Sep 14 2014
Formula
G.f.: (1-z-sqrt(1-2*z-3*z^2))/(2*(1-z)^2*sqrt(1-2*z-3*z^2)).
a(n) ~ 3^(n+3/2) / (8*sqrt(Pi*n)). - Vaclav Kotesovec, Sep 16 2014
D-finite with recurrence n*a(n) +(-5*n+3)*a(n-1) +2*(3*n-4)*a(n-2) +2*(n-1)*a(n-3) +(-7*n+16)*a(n-4) +3*(n-3)*a(n-5)=0. - R. J. Mathar, Jul 24 2022
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