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%I #49 Mar 09 2017 05:25:15
%S 1,1,3,6,16,38,100,254,674,1772,4760,12783,34745,94692,260040,716546,
%T 1984984,5517179,15396331,43094834,121008580,340686763,961686971,
%U 2720893669,7715273753,21921047638,62401862460,177948692666,508289340032,1454107965549
%N Number of Grand Dyck-Motzkin paths of length n.
%C 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.
%H G. C. Greubel, <a href="/A256943/b256943.txt">Table of n, a(n) for n = 0..1000</a>
%H L. Ferrari and E. Munarini, <a href="https://arxiv.org/abs/1203.6792"> Enumeration of edges in some lattices of paths </a>, arXiv:1203.6792 [math.CO], 2012.
%F 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.
%F a(n) ~ (3+sqrt(5)) * 3^(n+3/2) / (4*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Apr 20 2015
%e For instance, for n=3, we have the 6 paths UDH, HUD, HDU, DUH, DHU, HHH.
%t 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 *)
%o (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
%Y Cf. A002426.
%K nonn
%O 0,3
%A _José Luis Ramírez Ramírez_, Apr 19 2015
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