A325928 Number of Motzkin excursions of length n with an odd number of humps and an odd number of peaks.

0, 0, 1, 2, 4, 8, 17, 36, 83, 202, 519, 1382, 3766, 10352, 28551, 78756, 217224, 599542, 1657983, 4598766, 12803044, 35785664, 100412731, 282753476, 798690091, 2262087814, 6421507153, 18265543282, 52047980674, 148554917816, 424656556001, 1215691192244

Offset

0,4

Comments

A Motzkin excursion is a lattice path with steps from the set {D=-1, H=0, U=1} that starts at (0,0), never goes below the x-axis, and terminates at the altitude 0.

A peak is an occurrence of the pattern UD.

A hump is an occurrence of the pattern UHH...HD (the number of Hs in the pattern is not fixed, and can be 0).

Links

Table of n, a(n) for n=0..31.

Andrei Asinowski, Axel Bacher, Cyril Banderier, Bernhard Gittenberger, Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata, Algorithmica (2019).

Formula

G.f.: -1/2 + ( -sqrt((1-t)^2*(1+t)*(1-3*t)) + sqrt((1-2*t)*(1+t+2*t^2)*(1-t)^3) + sqrt((1+t^2)*(1-4*t+5*t^2)) - sqrt((1-2*t)*(1-2*t-t^2)*(1-t^2+2*t^3)) ) / (8*t^2*(1-t))

Example

For n=4, the a(4)=4 paths are UDHH, HUDH, HHUD, and UUDD (1 hump, 1 peak).

Prog

(PARI) seq(n)={my(t='x + O('x*'x^n)); Vec(-1/2 + ( -sqrt((1-t)^2*(1+t)*(1-3*t)) + sqrt((1-2*t)*(1+t+2*t^2)*(1-t)^3) + sqrt((1+t^2)*(1-4*t+5*t^2)) - sqrt((1-2*t)*(1-2*t-t^2)*(1-t^2+2*t^3)) ) / (8*t^2*(1-t)), -n)} \\ Andrew Howroyd, Aug 12 2019

Crossrefs

Motzkin meanders and excursions with parity restrictions on the number of humps and peaks:

A325921: Meanders, #humps=EVEN, #peaks=EVEN.

A325922: Excursions, #humps=EVEN, #peaks=EVEN.

A325923: Meanders, #humps=ODD, #peaks=EVEN.

A325924: Excursions, #humps=ODD, #peaks=EVEN.

A325925: Meanders, #humps=EVEN, #peaks=ODD.

A325926: Excursions, #humps=EVEN, #peaks=ODD.

A325927: Meanders, #humps=ODD, #peaks=ODD.

A325928 (this sequence): Excursions, #humps=ODD, #peaks=ODD.

Sequence in context: A093951 A137255 A247298 * A076892 A106462 A129987

Adjacent sequences: A325925 A325926 A325927 * A325929 A325930 A325931

Keyword

nonn

Author

Andrei Asinowski, Aug 10 2019

Status

approved