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A325928 Number of Motzkin excursions of length n with an odd number of humps and an odd number of peaks. 4
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 (list; graph; refs; listen; history; text; internal format)
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 restictions 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

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Last modified October 23 23:51 EDT 2019. Contains 328379 sequences. (Running on oeis4.)