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Search: id:A126222
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| A126222 |
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Triangle read by rows: T(n,k) is the number of 2-Motzkin paths (i.e. Motzkin paths with blue and red level steps) without red level steps on the x-axis, having length n and k level steps (0<=k<=n). |
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+0
2
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| 1, 0, 1, 1, 0, 1, 0, 4, 0, 1, 2, 0, 11, 0, 1, 0, 15, 0, 26, 0, 1, 5, 0, 69, 0, 57, 0, 1, 0, 56, 0, 252, 0, 120, 0, 1, 14, 0, 364, 0, 804, 0, 247, 0, 1, 0, 210, 0, 1800, 0, 2349, 0, 502, 0, 1, 42, 0, 1770, 0, 7515, 0, 6455, 0, 1013, 0, 1, 0, 792, 0, 11055, 0, 27940, 0, 16962, 0
(list; table; graph; listen)
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OFFSET
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0,8
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COMMENT
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Row sums are the Catalan numbers (A000108). T(2n,0)=C(2n,n)/(n+1) (the Catalan numbers; A000108). Sum(k*T(n,k), k=0..n)=A126223(n).
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FORMULA
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G.f.=G=G(t,z) satisfies z(t+z-t^2*z)G^2-G+1=0.
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EXAMPLE
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T(3,1)=4 because we have BUD, UBD, URD and UDB, where U=(1,1), D=(1,-1), B=blue (1,0), R=red (1,0).
Triangle starts:
1;
0,1;
1,0,1;
0,4,0,1;
2,0,11,0,1;
0,15,0,26,0,1;
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MAPLE
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G:=(1-sqrt(1-4*z*t-4*z^2+4*z^2*t^2))/2/z/(t+z-t^2*z): Gser:=simplify(series(G, z=0, 15)): for n from 0 to 12 do P[n]:=sort(expand(coeff(Gser, z, n))) od: for n from 0 to 12 do seq(coeff(P[n], t, j), j=0..n) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000108, A126223.
Sequence in context: A061309 A059064 A096459 this_sequence A071637 A141277 A096793
Adjacent sequences: A126219 A126220 A126221 this_sequence A126223 A126224 A126225
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 28 2006
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