

A329686


Number of excursions of length n with Motzkinsteps 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
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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 xaxis and never crossing the xaxis, i.e., staying at nonnegative altitude.


LINKS

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


FORMULA

G.f.: ((1+t)(1+t2t^4(1+t)*sqrt(14t^4)))/(2t^5).
a(4n)=2*C(n), a(4n1)=C(n), a(4n+1)=C(n) and a(4n+2)=0, where C(n) are the Catalan numbers A000108.
Dfinite with recurrence: (n+5)*(3*n^227*n+92)*a(n) +16*(3*n19)*a(n1) +16*(3*n+22)*a(n2) +16*(3*n25)*a(n3) 4*(n3)*(3*n^221*n+68)*a(n4)=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



