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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).).

%I A109189 #8 Feb 05 2014 17:42:45
%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,
%T A109189 128,72,56,10,12,0,1,310,324,254,112,80,12,14,0,1,822,932,654,432,160,
%U A109189 108,14,16,0,1,2238,2540,1986,1128,670,216,140,16,18,0,1,6094,7164,5546
%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).
%H A109189 Alois P. Heinz, <a href="/A109189/b109189.txt">Rows n = 0..140, flattened</a>
%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;
%p A109189 # second Maple program:
%p A109189 b:= proc(x, y) option remember;
%p A109189       `if`(abs(y)>x, 0, `if`(x=0, 1, expand(b(x-1, y)*
%p A109189       `if`(y=0, t, 1) +b(x-1, y+1) +b(x-1, y-1))))
%p A109189     end:
%p A109189 T:= n-> (p-> seq(coeff(p, t, i), i=0..n))(b(n, 0)):
%p A109189 seq(T(n), n=0..10);  # _Alois P. Heinz_, Feb 05 2014
%t A109189 nn=10;m=(1-x-(1-2x-3x^2)^(1/2))/(2x^2);CoefficientList[Series[1/(1-y x-2x^2m),{x,0,nn}],{x,y}]//Grid (* _Geoffrey Critzer_, Feb 05 2014 *)
%Y A109189 Cf. A002426, A109190, A015518, A001006.
%K A109189 nonn,tabl
%O A109189 0,4
%A A109189 _Emeric Deutsch_, Jun 21 2005