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Search: id:A126218
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| A126218 |
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Triangle read by rows: T(n,k) is the number of 0-1-2 trees (i.e. ordered trees with all vertices of outdegree at most two) with n edges and k pairs of adjacent vertices of outdegree 2. |
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+0
1
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| 1, 1, 2, 4, 7, 2, 13, 8, 26, 20, 5, 52, 50, 25, 104, 130, 75, 14, 212, 322, 217, 84, 438, 770, 644, 294, 42, 910, 1836, 1806, 952, 294, 1903, 4362, 4830, 3108, 1176, 132, 4009, 10268, 12738, 9576, 4188, 1056, 8494, 24032, 33219, 27948, 14760, 4752, 429, 18080
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OFFSET
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0,3
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COMMENT
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Row n has floor(n/2) terms (n>=2). Row sums are the Motzkin numbers (A001006). T(n,1)=A023431(n+1). Sum(k*T(n,k),k=0..floor(n/2)-1)=2*A014532(n-3) (n>=4).
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FORMULA
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G.f.=G=G(t,z) satisfies G=1+zG+z^2*[1+zG+t(G-1-zG)]^2 (see the Maple program for the explicit expression).
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EXAMPLE
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Triangle starts:
1;
1;
2;
4;
7,2;
13,8;
26,20,5;
52,50,25;
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MAPLE
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G:=1/2*(2*z^2*t^2-z+4*z^3*t-2*z^3*t^2-2*z^2*t-2*z^3+1-sqrt(1+4*z^3*t-4*z^2*t+z^2-2*z-4*z^3))/z^2/(z*t-t-z)^2: Gser:=simplify(series(G, z=0, 18)): for n from 0 to 15 do P[n]:=sort(coeff(Gser, z, n)) od: 1; 1; for n from 2 to 15 do seq(coeff(P[n], t, j), j=0..floor(n/2)-1) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A001006, A023431, A014532.
Sequence in context: A118390 A134974 A133292 this_sequence A086330 A098283 A137282
Adjacent sequences: A126215 A126216 A126217 this_sequence A126219 A126220 A126221
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 24 2006
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