%I A109189
%S A109189 1,0,1,2,0,1,2,4,0,1,8,4,6,0,1,16,20,6,8,0,1,46,40,36,8,10,0,1,114,128,
%T A109189 72,56,10,12,0,1,310,324,254,112,80,12,14,0,1,822,932,654,432,160,108,
%U A109189 14,16,0,1,2238,2540,1986,1128,670,216,140,16,18,0,1,6094,7164,5546
%N A109189 Triangle read by rows: T(n,k) is number of Grand Motzkin paths of length 
               n having k (1,0)-steps at level zero. (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).).
%C A109189 Row sums yield the central trinomial coefficients (A002426). T(n,0)=A109190(n). 
               sum(k*T(n,k),k=0..n)=A015518(n).
%F A109189 G.f.= 1/(1-tz-2z^2*M), where M=1+zM+z^2*M^2=[1-z-sqrt(1-2z-3z^2)]/(2z^2) 
               is the g.f. of the Motzkin numbers (A001006).
%e A109189 T(4,1)=4 because we have (h)uhd, (h)dhu, uhd(h) and dhu(h), where u=(1,
               1),d=(1,-1), h=(1,0) and the (1,0) steps at level 0 are shown between 
               parentheses.
%e A109189 Triangle begins:
%e A109189 1;
%e A109189 0,1;
%e A109189 2,0,1;
%e A109189 2,4,0,1;
%e A109189 8,4,6,0,1;
%e A109189 16,20,6,8,0,1;
%p A109189 M:=(1-z-sqrt(1-2*z-3*z^2))/2/z^2: G:=1/(1-t*z-2*z^2*M): Gser:=simplify(series(G,
               z=0,13)): P[0]:=1: for n from 1 to 11 do P[n]:=coeff(Gser,z^n) od: 
               for n from 0 to 11 do seq(coeff(t*P[n],t^k),k=1..n+1) od;
%Y A109189 Cf. A002426, A109190, A015518, A001006.
%Y A109189 Sequence in context: A115346 A140531 A117316 this_sequence A144172 A166692 
               A046766
%Y A109189 Adjacent sequences: A109186 A109187 A109188 this_sequence A109190 A109191 
               A109192
%K A109189 nonn,tabl
%O A109189 0,4
%A A109189 Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 21 2005