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A109195 Triangle read by rows: T(n,k) is number of Grand Motzkin paths of length n having k returns to the x-axis from above (i.e., d steps hitting the x-axis).

%I A109195 #21 Jul 27 2017 04:19:23
%S A109195 1,1,2,1,4,3,9,9,1,21,25,5,51,69,20,1,127,189,70,7,323,518,230,35,1,
%T A109195 835,1422,726,147,9,2188,3915,2235,560,54,1,5798,10813,6765,2002,264,
%U A109195 11,15511,29964,20240,6853,1143,77,1,41835,83304,60060,22737,4563,429,13
%N A109195 Triangle read by rows: T(n,k) is number of Grand Motzkin paths of length n having k returns to the x-axis from above (i.e., d steps hitting the x-axis).
%C A109195 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 A109195 Row n contains 1 + floor(n/2) terms. Row sums yield the central trinomial coefficients (A002426).
%C A109195 Column k is the sum of columns 2k and 2k+1 of A089942. - _Philippe Deléham_, Nov 11 2008
%F A109195 T(n,0) = A001006(n) (the Motzkin numbers).
%F A109195 Sum_{k=0..floor(n/2)} k*T(n,k) = A109196(n).
%F A109195 G.f.: 1/(1 - z - (1+t)z^2*M), where M = 1 + zM + z^2*M^2 = (1 - z - sqrt(1 - 2z - 3z^2))/(2z^2) is the g.f. for the Motzkin numbers (A001006).
%F A109195 T(n,k) = A089942(n,2*k) + A089942(n,2*k+1). - _Philippe Deléham_, Nov 11 2008
%e A109195 T(3,1)=3 because we have hud, udh and uhd, where u=(1,1),d=(1,-1), h=(1,0).
%e A109195 Triangle begins:
%e A109195    1;
%e A109195    1;
%e A109195    2,  1;
%e A109195    4,  3;
%e A109195    9,  9,  1;
%e A109195   21, 25,  5;
%e A109195   51, 69, 20,  1;
%p A109195 M:=(1-z-sqrt(1-2*z-3*z^2))/2/z^2: G:=1/(1-z-(1+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 A109195 Cf. A001006, A002426, A109196.
%K A109195 nonn,tabf
%O A109195 0,3
%A A109195 _Emeric Deutsch_, Jun 22 2005