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A109188 Number of (1,0) steps in all Grand Motzkin paths of length n.

%I A109188 #31 Nov 09 2021 13:12:54
%S A109188 1,2,9,28,95,306,987,3144,9963,31390,98483,307836,959257,2981174,
%T A109188 9243405,28601712,88342659,272428758,838903371,2579937060,7924966749,
%U A109188 24317716038,74546117121,228317474952,698708409525,2136597743826
%N A109188 Number of (1,0) steps in all Grand Motzkin paths of length n.
%C A109188 A Grand 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).
%H A109188 G. C. Greubel, <a href="/A109188/b109188.txt">Table of n, a(n) for n = 1..1000</a>
%F A109188 G.f.: x*(1 - x)/(1 - 2*x - 3*x^2)^(3/2).
%F A109188 a(n) = n*A002426(n-1). - _Paul Barry_, Apr 19 2008, corrected Nov 09 2021
%F A109188 E.g.f.: a(n) = n! * [x^n] exp(x)*((1 + x)*BesselI(0, 2*x) + 2*x*BesselI(1, 2*x)). - _Peter Luschny_, Aug 25 2012
%F A109188 D-finite with recurrence (-n+1)*a(n) + (3*n-4)*a(n-1) + (n+5)*a(n-2) + 3*(-n+2)*a(n-3) = 0. - _R. J. Mathar_, Nov 26 2012
%F A109188 a(n) = n*hypergeom([1-n/2, 1/2-n/2], [1], 4) . - _Peter Luschny_, Sep 18 2014
%F A109188 a(n) ~ 3^(n-1/2)*sqrt(n)/(2*sqrt(Pi)). - _Vaclav Kotesovec_, Sep 18 2014
%e A109188 a(3)=9 because we have the following 7 (=A002426(3)) Grand Motzkin paths of length 3: hhh, hud, hdu, udh, duh, uhd and dhu; they have a total of 9 h-steps.
%p A109188 g:=z*(1-z)/(1-2*z-3*z^2)^(3/2): gser:=series(g,z=0,33): seq(coeff(gser,z^n),n=1..30);
%p A109188 a := n -> n*hypergeom([1-n/2, 1/2-n/2], [1], 4):
%p A109188 seq(simplify(a(n)), n=1..26); # _Peter Luschny_, Sep 18 2014
%t A109188 Rest[CoefficientList[Series[x*(1-x)/(1-2*x-3*x^2)^(3/2), {x, 0, 20}], x]] (* _Vaclav Kotesovec_, Sep 18 2014 *)
%o A109188 (PARI) Vec(z*(1-z)/(1-2*z-3*z^2)^(3/2) + O(z^50)) \\ _G. C. Greubel_, Jan 31 2017
%Y A109188 Cf. A109187, A002426.
%K A109188 nonn
%O A109188 1,2
%A A109188 _Emeric Deutsch_, Jun 21 2005