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A256943 Number of Grand Dyck-Motzkin paths of length n. 1
1, 1, 3, 6, 16, 38, 100, 254, 674, 1772, 4760, 12783, 34745, 94692, 260040, 716546, 1984984, 5517179, 15396331, 43094834, 121008580, 340686763, 961686971, 2720893669, 7715273753, 21921047638, 62401862460, 177948692666, 508289340032, 1454107965549 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A Grand Dyck-Motzkin path is a path in the half-plane x>=0, starting at (0,0), ending at (n,0) and consisting of steps U=(1,1), D=(1,-1) and H=(1,0), such that H-steps are only allowed if y<=0.

LINKS

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

L. Ferrari and E. Munarini, Enumeration of edges in some lattices of paths , arXiv:1203.6792 [math.CO], 2012.

FORMULA

G.f.: 1/(1-x-x^2*C(x^2)-x^2*M(x)), where C(x) is the g.f. of Catalan numbers and M(x) is the g.f. of Motzkin paths.

a(n) ~ (3+sqrt(5)) * 3^(n+3/2) / (4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 20 2015

EXAMPLE

For instance, for n=3, we have the 6 paths UDH, HUD, HDU, DUH, DHU, HHH.

MATHEMATICA

CoefficientList[Series[2/(Sqrt[1-4*x^2] + Sqrt[1-2*x-3*x^2] - x), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 20 2015 *)

PROG

(PARI) x='x+O('x^50); Vec(2/(sqrt(1-4*x^2) + sqrt(1-2*x-3*x^2) - x)) \\ G. C. Greubel, Mar 09 2017

CROSSREFS

Cf. A002426.

Sequence in context: A114410 A190735 A096588 * A275207 A073079 A143560

Adjacent sequences:  A256940 A256941 A256942 * A256944 A256945 A256946

KEYWORD

nonn

AUTHOR

José Luis Ramírez Ramírez, Apr 19 2015

STATUS

approved

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Last modified July 15 04:34 EDT 2020. Contains 335763 sequences. (Running on oeis4.)