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A329686 Number of excursions of length n with Motzkin-steps forbidding all consecutive steps of length 2 except UH, HU, HD and DH. 0
1, 1, 0, 1, 2, 1, 0, 2, 4, 2, 0, 5, 10, 5, 0, 14, 28, 14, 0, 42, 84, 42, 0, 132, 264, 132, 0, 429, 858, 429, 0, 1430, 2860, 1430, 0, 4862, 9724, 4862, 0, 16796, 33592, 16796, 0, 58786, 117572, 58786, 0, 208012, 416024, 208012, 0, 742900, 1485800, 742900, 0, 2674440, 5348880, 2674440, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

The Motzkin step set is U=(1,1), H=(1,0) and D=(1,-1). An excursion is a path starting at (0,0), ending on the x-axis and never crossing the x-axis, i.e., staying at nonnegative altitude.

LINKS

Table of n, a(n) for n=0..58.

FORMULA

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

a(4n)=2*C(n), a(4n-1)=C(n), a(4n+1)=C(n) and a(4n+2)=0, where C(n) are the Catalan numbers A000108.

D-finite with recurrence: (n+5)*(3*n^2-27*n+92)*a(n) +16*(3*n-19)*a(n-1) +16*(-3*n+22)*a(n-2) +16*(3*n-25)*a(n-3) -4*(n-3)*(3*n^2-21*n+68)*a(n-4)=0. - R. J. Mathar, Jan 09 2020

EXAMPLE

a(8)=4 since we have the following four excursions of length 8: UHDHUHDH, HUHDHUHD, UHUHDHDH and HUHUHDHD.

CROSSREFS

Cf. A000108.

Sequence in context: A279226 A108354 A329099 * A261615 A146162 A147702

Adjacent sequences:  A329683 A329684 A329685 * A329687 A329688 A329689

KEYWORD

nonn,walk

AUTHOR

Valerie Roitner, Nov 29 2019

STATUS

approved

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Last modified September 23 14:40 EDT 2021. Contains 347618 sequences. (Running on oeis4.)