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A109193 Triangle read by rows: T(n,k) is number of Grand Motzkin paths of length n having k returns to the x-axis (i.e., d or u steps hitting the x-axis).

%I A109193 #10 Jul 27 2017 04:19:15
%S A109193 1,1,1,2,1,6,1,14,4,1,30,20,1,64,68,8,1,140,196,56,1,318,524,248,16,1,
%T A109193 750,1356,888,144,1,1828,3476,2832,784,32,1,4576,8932,8448,3344,352,1,
%U A109193 11700,23136,24248,12368,2272,64,1,30420,60528,68120,41808,11232,832,1
%N A109193 Triangle read by rows: T(n,k) is number of Grand Motzkin paths of length n having k returns to the x-axis (i.e., d or u steps hitting the x-axis).
%C A109193 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 A109193 Row n contains 1 + floor(n/2) terms.
%C A109193 Row sums yield the central trinomial coefficients (A002426).
%F A109193 T(n,0) = 1.
%F A109193 Sum_{k=0..floor(n/2)} k*T(n,k) = A109194(n).
%F A109193 G.f.: 1/(1 - z - 2tz^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 A109193 T(4,2)=4 because we have udud, dudu, uddu and duud, where u=(1,1), d=(1,-1), h=(1,0).
%e A109193 Triangle begins:
%e A109193   1;
%e A109193   1;
%e A109193   1,  2;
%e A109193   1,  6;
%e A109193   1, 14,  4;
%e A109193   1, 30, 20;
%e A109193   1, 64, 68,  8;
%p A109193 M:=(1-z-sqrt(1-2*z-3*z^2))/2/z^2: G:=1/(1-z-2*t*z^2*M): Gser:=simplify(series(G,z=0,17)): P[0]:=1: for n from 1 to 14 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 14 do seq(coeff(t*P[n],t^k),k=1..1+floor(n/2)) od; # yields sequence in triangular form
%Y A109193 Cf. A001006, A002426, A109194.
%K A109193 nonn,tabf
%O A109193 0,4
%A A109193 _Emeric Deutsch_, Jun 22 2005