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A109190 Number of (1,0)-steps at level zero in all Grand Motzkin paths of length n.

%I A109190 #40 Jul 17 2023 04:55:54
%S A109190 1,0,2,2,8,16,46,114,310,822,2238,6094,16764,46308,128650,358862,
%T A109190 1005056,2824416,7962122,22508350,63792424,181219680,515905018,
%U A109190 1471593638,4205280902,12037415526,34510499066,99083855234,284870069780
%N A109190 Number of (1,0)-steps at level zero in all Grand Motzkin paths of length n.
%C A109190 A Grand Motzkin path of length n is a path in the half-plane x>=0, starting at (0,0), ending at (n,0) with steps u=(1,1), d=(1,-1) and h=(1,0).
%C A109190 Column 0 of A109189.
%C A109190 The substitution x->x/(1+x+x^2) in the g.f. (this might be called an inverse Motzkin transform), yields the g.f. of (-1)^n*A006355(n). - _R. J. Mathar_, Nov 10 2008
%C A109190 Apparently also the number of grand Motzkin paths of length n that avoid flat steps at level 0. - _David Scambler_, Jul 04 2013
%C A109190 Motzkin contexts such that along the path from the root to the hole there are only binary nodes. - _Pierre Lescanne_, Nov 11 2015
%H A109190 Alois P. Heinz, <a href="/A109190/b109190.txt">Table of n, a(n) for n = 0..1000</a>
%H A109190 Paul Barry, <a href="https://arxiv.org/abs/2307.00098">Moment sequences, transformations, and Spidernet graphs</a>, arXiv:2307.00098 [math.CO], 2023.
%H A109190 Taras Goy and Mark Shattuck, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Shattuck/sh36.html">Determinants of Some Hessenberg-Toeplitz Matrices with Motzkin Number Entries</a>, J. Int. Seq., Vol. 26 (2023), Article 23.3.4.
%F A109190 G.f.: (sqrt(1-2*z-3*z^2)-z)/(1-2*z-4*z^2).
%F A109190 G.f.: 1/(1-2x^2*M(x)), M(x) the g.f. of the Motzkin numbers A001006. - _Paul Barry_, Mar 02 2010
%F A109190 D-finite with recurrence n*a(n) +(3-4*n)*a(n-1) +3*(1-n)*a(n-2) +2*(7*n-15)*a(n-3) +12*(n-3)*a(n-4) = 0. - _R. J. Mathar_, Nov 09 2012
%F A109190 a(n) ~ 3^(n+3/2) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Feb 03 2014
%e A109190 a(3) = 2 because we have uhd and dhu.
%p A109190 g:=(sqrt(1-2*z-3*z^2)-z)/(1-2*z-4*z^2): gser:=series(g,z=0,33): 1,seq(coeff(gser,z^n),n=1..30);
%t A109190 CoefficientList[Series[(Sqrt[1-2*x-3*x^2]-x)/(1-2*x-4*x^2), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 03 2014 *)
%o A109190 (PARI) x='x+O('x^55); Vec((sqrt(1-2*x-3*x^2)-x)/(1-2*x-4*x^2)) \\ _Altug Alkan_, Nov 11 2015
%Y A109190 Cf. A001006, A006355, A109189.
%K A109190 nonn
%O A109190 0,3
%A A109190 _Emeric Deutsch_, Jun 21 2005