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%I #6 Jul 26 2017 20:29:09
%S 1,1,2,1,5,2,13,5,1,34,14,3,91,40,9,1,247,114,28,4,678,327,87,14,1,
%T 1877,942,267,48,5,5233,2723,815,161,20,1,14674,7892,2478,528,75,6,
%U 41349,22924,7512,1706,270,27,1,117001,66712,22718,5452,941,110,7,332260
%N Triangle read by rows: T(n,k) is number of Grand Motzkin paths of length n having k hills (i.e., ud's starting at level 0). (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 Row n contains 1 + floor(n/2) terms.
%C Row sums yield the central trinomial coefficients (A002426).
%C T(n,0) = A109192(n).
%C Sum_{k=0..floor(n/2)} k*T(n,k) = A015518(n-1).
%F G.f.: 1/(1 - z + z^2 - tz^2 - 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 T(3,1)=2 because we have hud and udh, where u=(1,1),d=(1,-1), h=(1,0).
%e Triangle begins:
%e 1;
%e 1;
%e 2, 1;
%e 5, 2;
%e 13, 5, 1;
%e 34, 14, 3;
%e 91, 40, 9, 1;
%p M:=(1-z-sqrt(1-2*z-3*z^2))/2/z^2: G:=1/(1-z+z^2-t*z^2-2*z^2*M): Gser:=simplify(series(G,z=0,16)): 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;
%Y Cf. A001006, A002426, A015518, A109192.
%K nonn,tabf
%O 0,3
%A _Emeric Deutsch_, Jun 21 2005
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