A191398 Number of dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0)-steps at positive heights) having no DHU's (here U=(1,1), H=(1,0), and D=(1,-1)).

1, 1, 2, 3, 6, 9, 18, 28, 56, 89, 179, 289, 585, 956, 1948, 3214, 6591, 10959, 22609, 37833, 78486, 132037, 275316, 465255, 974659, 1653418, 3478520, 5920569, 12504448, 21344348, 45240473, 77417309, 164624203, 282335973, 602163830, 1034757445, 2212959172, 3809387953, 8167344875

Offset: 0

Author

Emeric Deutsch, Jun 04 2011

Keywords

• nonn

Examples

			a(5)=9 because among the 10 (=A001405(5)) dispersed Dyck paths of length 5 only UDHUD has a DHU.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Programs

Maple

g := 2/(1-z-2*z^3+(1-z)*sqrt(1-4*z^2)); gser := series(g, z = 0, 41); seq(coeff(gser, z, n), n = 0 .. 38);

Mathematica

CoefficientList[Series[2/(1-x-2*x^3+(1-x)*Sqrt[1-4*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 21 2014 *)

PARI

x='x+O('x^50); Vec(2/(1-x-2*x^3+(1-x)*sqrt(1-4*x^2))) \\ G. C. Greubel, Mar 26 2017

Formula

a(n) = A191397(n,0).

G.f.: 2/(1-z-2*z^3+(1-z)*sqrt(1-4*z^2)).

a(n) ~ 2^(n+7/2) * (1+3*(-1)^n/49) / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 21 2014

Conjecture: -(n+2)*(n-3)*a(n) +(3*n^2-3*n-14)*a(n-1) +2*(n^2-7*n+8)*a(n-2) +4*(-3*n^2+12*n-10)*a(n-3) +(7*n^2-31*n+38)*a(n-4) +4*a(n-5) +4*(n-2)^2*a(n-6)=0. - R. J. Mathar, Jun 14 2016