A089380 Number of Motzkin paths of length n with no UD, UHD, UHHD, UHHHD, ..., starting at level zero (here H=(1,0), U=(1,1), D=(1,-1)).

1, 1, 1, 1, 2, 6, 18, 50, 133, 349, 919, 2443, 6559, 17759, 48417, 132765, 365883, 1012827, 2814975, 7852359, 21977172, 61697208, 173688720, 490222392, 1386896799, 3932313671, 11172152779, 31801604227, 90683754826, 259017103918

Offset

0,5

Links

Table of n, a(n) for n=0..29.

Formula

G.f.=2(1-z)/[1-2z+3z^2+(1-z)sqrt(1-2z-3z^2)].

D-finite with recurrence 2*(n+2)*a(n) +2*(-5*n-4)*a(n-1) +(13*n+2)*a(n-2) +(n-16)*a(n-3) +3*(-5*n+14)*a(n-4) +9*(n-4)*a(n-5)=0. - R. J. Mathar, Jul 24 2022

Example

a(5)=6 because among the 21 Motzkin paths of length 5 only the following have
no U(H^p)D for any p>=0: HHHHH, HUUDD, UUDDH, UHUDD, UUDHD and UUHDD.

Crossrefs

Cf. A001006.

Sequence in context: A258625 A062026 A048495 * A271897 A362067 A304962

Adjacent sequences: A089377 A089378 A089379 * A089381 A089382 A089383

Keyword

nonn

Author

Emeric Deutsch, Dec 27 2003

Status

approved