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%I #20 Jul 26 2022 14:27:34
%S 2,6,22,70,224,700,2174,6702,20572,62920,191932,584220,1775258,
%T 5386846,16326734,49435150,149557436,452133880,1366012832,4124825872,
%U 12449394278,37558361290,113266431860,341467468420,1029119688014
%N Number of returns to the x-axis (i.e., d or u steps hitting the x-axis) in all Grand Motzkin paths of length n. (A Grand Motzkin path of length n 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 G. C. Greubel, <a href="/A109194/b109194.txt">Table of n, a(n) for n = 2..1000</a>
%F a(n) = Sum_{k=0..floor(n/2)} k*A109193(n,k).
%F a(n) = 2*A109196(n).
%F G.f.: (1-z-sqrt(1-2*z-3*z^2))/(1-2*z-3*z^2).
%F a(n) = 2*Sum_(k=1..n} ( Sum_{j=0..n} binomial(j,-n-2*k+2*j)*binomial(n,j)) ), n>1. - _Vladimir Kruchinin_, Oct 12 2011
%F a(n) ~ 3^n/2 * (1-sqrt(3/(Pi*n))). - _Vaclav Kotesovec_, Nov 05 2016
%F D-finite with recurrence n*a(n) +(-4*n+3)*a(n-1) +(-2*n+3)*a(n-2) +3*(4*n-9)*a(n-3) +9*(n-3)*a(n-4)=0. - _R. J. Mathar_, Jul 26 2022
%e a(3)=6 because we have the following 7 (=A002426(3)) Grand Motzkin paths of length 3: hhh, hu(d), hd(u), u(d)h, d(u)h, uh(d) and dh(u); they have a total of 6 returns to the x-axis (shown between parentheses).
%p g:=(1-z-sqrt(1-2*z-3*z^2))/(1-2*z-3*z^2): gser:=series(g,z=0,30): seq(coeff(gser,z^n),n=2..28);
%t Drop[CoefficientList[Series[(1-x-Sqrt[1-2*x-3*x^2])/(1-2*x-3*x^2), {x, 0, 30}], x], 2] (* _Vaclav Kotesovec_, Nov 05 2016 *)
%o (PARI) x='x+O('x^50); Vec((1-x-sqrt(1-2*x-3*x^2))/(1-2*x-3*x^2)) \\ _G. C. Greubel_, Mar 02 2017
%Y Cf. A109193, A109196.
%K nonn
%O 2,1
%A _Emeric Deutsch_, Jun 22 2005
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