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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). (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).). 2
 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, 835, 1422, 726, 147, 9, 2188, 3915, 2235, 560, 54, 1, 5798, 10813, 6765, 2002, 264, 11, 15511, 29964, 20240, 6853, 1143, 77, 1, 41835, 83304, 60060, 22737, 4563, 429, 13 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Row n contains 1+floor(n/2) terms. Row sums yield the central trinomial coefficients (A002426). T(n,0)=A001006(n) (the Motzkin numbers). sum(k*T(n,k),k=0..floor(n/2))=A109196(n). Column k is the sum of columns 2k and 2k+1 of A089942. [From Philippe DELEHAM, Nov 11 2008] LINKS FORMULA 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). T(n,k)=A089942(n,2k)+A089942(n,2k+1). [From Philippe DELEHAM, Nov 11 2008] EXAMPLE T(3,1)=3 because we have hud, udh and uhd, where u=(1,1),d=(1,-1), h=(1,0). Triangle begins: 1; 1; 2,1; 4,3; 9,9,1; 21,25,5; 51,69,20,1; MAPLE 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 CROSSREFS Cf. A002426, A001006, A109196. Sequence in context: A207537 A114438 A181882 * A217927 A032662 A185413 Adjacent sequences:  A109192 A109193 A109194 * A109196 A109197 A109198 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Jun 22 2005 STATUS approved

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