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A247289 Number of weak peaks in all peakless Motzkin paths of length n. 2
0, 0, 0, 2, 7, 18, 45, 110, 267, 652, 1602, 3960, 9845, 24594, 61689, 155270, 391962, 991968, 2515964, 6393610, 16275174, 41491776, 105922244, 270734244, 692756227, 1774418286, 4549173861, 11672860634, 29975156134, 77029918152, 198083586300, 509692521982 (list; graph; refs; listen; history; text; internal format)
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).

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

Cf. A004148, A247288.

Sequence in context: A076857 A243717 A174192 * A161870 A072338 A182197

Adjacent sequences:  A247286 A247287 A247288 * A247290 A247291 A247292

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Sep 14 2014

STATUS

approved

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Last modified November 14 20:15 EST 2019. Contains 329130 sequences. (Running on oeis4.)