OFFSET
0,4
COMMENTS
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 peakless Motzkin path uhu*h*ddu*h*h*d, where u=(1,1), h=(1,0), d(1,-1), has 5 weak peaks (shown by the stars).
a(n) = Sum(k*A247288(n,k), 0<=k<=n-1).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: (2-z)*z^3*g/((1-z)^2*(1-z+z^2-2*z^2*g)), where g is defined by g = 1 + z*g + z^2*g*(g-1).
D-finite with recurrence n*(n-1)*a(n) +(-7*n^2+28*n-31)*a(n-1) +(n-2)*(13*n-48)*a(n-2) +(-5*n^2+21*n-6)*a(n-3) +(7*n^2-43*n+82)*a(n-4) -(13*n-24)*(n-5)*a(n-5) +(4*n-5)*(n-6)*a(n-6)=0. - R. J. Mathar, Jul 24 2022
EXAMPLE
a(4)=7 because the peakless Motzkin paths u*h*dhh, hu*h*dh, and u*h*h*d have 0, 2, 2, and 3 weak peaks (shown by the stars).
MAPLE
f := (2-z)*z^3*g/((1-z)^2*(1-z+z^2-2*z^2*g)): eqg := g = 1+z*g+z^2*g*(g-1): g := RootOf(eqg, g): fser := series(f, z = 0, 35): seq(coeff(fser, z, n), n = 0 .. 33);
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Sep 14 2014
STATUS
approved