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

0,4

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

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

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