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A325927 Number of Motzkin meanders of length n with an odd number of humps and an odd number of peaks. 4
0, 0, 1, 4, 13, 38, 105, 280, 737, 1942, 5183, 14100, 39151, 110642, 316751, 914248, 2650655, 7701562, 22400559, 65203428, 189970159, 554165922, 1619018259, 4737859512, 13887657307, 40769959314, 119849273449, 352716050428, 1039027117929 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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

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..28.

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

EXAMPLE

For n=3, the a(3)=4 paths are UDH, UDU, UUD, HUD (1 hump, 1 peak).

PROG

(PARI) seq(n)={my(t='x + O('x*'x^n)); Vec(( sqrt((1+t)/(1-3*t)) - sqrt((1+t+2*t^2)/((1-2*t)*(1-t))) - sqrt((1+t^2)/(1-4*t+5*t^2)) + sqrt((1-t^2+2*t^3)/((1-2*t)*(1-t^2-2*t))) ) / (8*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  (this sequence): Meanders, #humps=ODD, #peaks=ODD.

A325928: Excursions, #humps=ODD, #peaks=ODD.

Sequence in context: A049611 A084851 A094706 * A056014 A247287 A159036

Adjacent sequences:  A325924 A325925 A325926 * A325928 A325929 A325930

KEYWORD

nonn

AUTHOR

Andrei Asinowski, Aug 10 2019

STATUS

approved

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Last modified June 13 09:02 EDT 2021. Contains 344981 sequences. (Running on oeis4.)