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

0,4

COMMENTS

a(n) = Sum(k*A247286(n,k), 0<=k<=n).

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

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

EXAMPLE

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).

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);

PROG

(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

CROSSREFS

Cf. A001006, A247286.

Sequence in context: A094706 A325927 A056014 * A159036 A058693 A027076

Adjacent sequences:  A247284 A247285 A247286 * A247288 A247289 A247290

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Sep 14 2014

STATUS

approved

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Last modified September 17 01:53 EDT 2021. Contains 347478 sequences. (Running on oeis4.)