A190171 Number of peakless Motzkin paths of length n having no UHD's starting at level 0; here U=(1,1), H=(1,0), and D=(1,-1).

1, 1, 1, 1, 2, 5, 12, 27, 60, 135, 309, 717, 1680, 3966, 9423, 22518, 54091, 130540, 316358, 769577, 1878497, 4599623, 11294640, 27807381, 68627188, 169746823, 420732391, 1044830875, 2599352149, 6477571270, 16167429874, 40411920571, 101153167258, 253522241008

Offset

0,5

Comments

a(n)=A190170(n,0).

Links

Table of n, a(n) for n=0..33.

Formula

G.f. G=G(z) is obtained by elimitaing S from the equations G=1+zG+z^2*G(S-1-z) and S=1+zS+z^2*S(S-1).

From Vaclav Kotesovec, May 29 2022: (Start)

G.f.: 2/(1 + sqrt((1 + (-3 + x)*x)*(1 + x + x^2)) + x*(-1 + x + 2*x^2)).

a(n) ~ 5^(1/4) * phi^(2*n+6) / (18 * sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. (End)

Conjecture D-finite with recurrence (n+2)*a(n) +(-n+1)*a(n-1) +2*(-2*n-1)*a(n-2) +9*a(n-3) +(-n+1)*a(n-4) -9*a(n-5) +2*(-2*n+5)*a(n-6) +(-n+1)*a(n-7) +(n-4)*a(n-8)=0. - R. J. Mathar, Jul 22 2022

Example

a(4)=2 because we have HHHH and UHHD.

Maple

p1 := G-1-z*G-z^2*G*(S-1-z): p2 := S-1-z*S-z^2*S*(S-1): r := resultant(p1, p2, S): g := RootOf(r, G): Gser := simplify(series(g, z = 0, 40)): seq(coeff(Gser, z, n), n = 0 .. 33);

Mathematica

CoefficientList[Series[2/(1 + Sqrt[(1 + (-3 + x)*x)*(1 + x + x^2)] + x*(-1 + x + 2*x^2)), {x, 0, 40}], x] (* Vaclav Kotesovec, May 29 2022 *)

Crossrefs

Cf. A190170.

Sequence in context: A304175 A086589 A299270 * A091596 A077863 A319172

Adjacent sequences: A190168 A190169 A190170 * A190172 A190173 A190174

Keyword

nonn

Author

Emeric Deutsch, May 06 2011

Status

approved