%I A109194
%S A109194 2,6,22,70,224,700,2174,6702,20572,62920,191932,584220,1775258,5386846,
%T A109194 16326734,49435150,149557436,452133880,1366012832,4124825872,
%U A109194 12449394278,37558361290,113266431860,341467468420,1029119688014
%N A109194 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).).
%C A109194 a(n)=sum(k*A109193(n,k),k=0..floor(n/2)). a(n)=2*A109196(n).
%F A109194 G.f.=[1-z-sqrt(1-2z-3z^2)]/(1-2z-3z^2).
%e A109194 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 A109194 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);
%Y A109194 Cf. A109193, A109196.
%Y A109194 Sequence in context: A027561 A126171 A002839 this_sequence A014334 A107239 
               A148496
%Y A109194 Adjacent sequences: A109191 A109192 A109193 this_sequence A109195 A109196 
               A109197
%K A109194 nonn
%O A109194 2,1
%A A109194 Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 22 2005