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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). 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 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). Row n contains 1 + floor(n/2) terms. Row sums yield the central trinomial coefficients (A002426). Column k is the sum of columns 2k and 2k+1 of A089942. - Philippe Deléham, Nov 11 2008 LINKS FORMULA T(n,0) = A001006(n) (the Motzkin numbers). Sum_{k=0..floor(n/2)} k*T(n,k) = A109196(n). 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,2*k) + A089942(n,2*k+1). - Philippe Deléham, 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. A001006, A002426, A109196. Sequence in context: A262230 A262155 A181882 * A258310 A217927 A284709 Adjacent sequences:  A109192 A109193 A109194 * A109196 A109197 A109198 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Jun 22 2005 STATUS approved

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Last modified October 23 18:52 EDT 2018. Contains 316530 sequences. (Running on oeis4.)