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Search: id:A126188
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| A126188 |
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Triangle read by rows: T(n,k) is the number hex trees with n edges and k pairs of adjacent vertices of outdegree 2. A hex tree is a rooted tree where each vertex has 0, 1, or 2 children and, when only one child is present, it is either a left child, or a middle child, or a right child (name due to an obvious bijection with certain tree-like polyhexes; see the Harary-Read reference). |
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+0
3
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| 1, 3, 10, 36, 135, 2, 519, 24, 2034, 180, 5, 8100, 1110, 75, 32688, 6210, 675, 14, 133380, 32886, 4851, 252, 549342, 168210, 30996, 2646, 42, 2280690, 840132, 184842, 21672, 882, 9534591, 4124682, 1053486, 154980, 10584, 132, 40103019
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Row n has floor(n/2) terms (n>=2). Sum of terms in row n = A002212(n+1). T(n,0)=A126189(n). Sum(k*T(n,k),k=0..floor(n/2)-1)=A126190(n).
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REFERENCES
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F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.
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FORMULA
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G.f.=G=G(t,z)=G=1+3zG+z^2*[1+3zG+t(G-1-3zG)]^2 (explicit expression in the Maple program).
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EXAMPLE
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Triangle starts:
1;
3;
10;
36;
135,2;
519,24;
2034,180,5;
8100,1110,75;
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MAPLE
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G:=1/2*(12*z^3*t+2*z^2*t^2-2*z^2*t-6*z^3*t^2-3*z-6*z^3+1-sqrt(1+9*z^2-4*z^2*t-6*z+12*z^3*t-12*z^3))/z^2/(3*z*t-t-3*z)^2: Gser:=simplify(series(G, z=0, 18)): for n from 0 to 14 do P[n]:=sort(coeff(Gser, z, n)) od: 1; 3; for n from 2 to 14 do seq(coeff(P[n], t, j), j=0..floor(n/2)-1) od;
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CROSSREFS
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Cf. A002212, A126189, A126190.
Sequence in context: A055989 A102871 A119374 this_sequence A081909 A126189 A122448
Adjacent sequences: A126185 A126186 A126187 this_sequence A126189 A126190 A126191
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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 25 2006
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