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Search: id:A126321
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| A126321 |
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Triangle read by rows: number of hex trees with n edges and k branches of length 1 (0<=k<=n). 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
4
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| 1, 0, 3, 9, 0, 1, 27, 6, 0, 3, 90, 18, 27, 0, 2, 297, 135, 81, 24, 0, 6, 1053, 648, 351, 72, 90, 0, 5, 3888, 2889, 1377, 756, 270, 90, 0, 15, 14742, 12150, 6723, 3888, 1485, 270, 315, 0, 14, 56619, 51273, 32805, 19224, 6480, 3645, 945, 336, 0, 42, 219429, 218700
(list; table; graph; listen)
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OFFSET
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0,3
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COMMENT
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Sum of terms in row n = A002212(n+1). T(n,0)=A126322(n). T(n,n)=A126324(n). T(2n,2n)=Cat(n); T(2n+1,2n+1)=3*Cat(n), where Cat(n) are the Catalan numbers (A000108). Sum(k*T(n,k),k=0..n)=A126323(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.=(1+P)C(z^2*Q^2), where C(z)=(1-sqrt(1-4z))/(2z) is the Catalan function, P=3tz + 9z^2/(1-3z) and Q=t+3z/(1-3z).
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EXAMPLE
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T(2,2)=1 because among the 10 hex trees with two edges only the tree V has 2 branches of length 1.
Triangle starts:
1;
0,3;
9,0,1;
27,6,0,3;
90,18,27,0,2;
297,135,81,24,0,6;
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MAPLE
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C:=z->(1-sqrt(1-4*z))/2/z: G:=(1+3*t*z+9*z^2/(1-3*z))*C(z^2*(t+3*z/(1-3*z))^2): Gser:=simplify(series(G, z=0, 15)): for n from 0 to 10 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 10 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. A002212, A126322, A126323, A126324, A000108.
Sequence in context: A088110 A122759 A016626 this_sequence A021260 A134878 A154540
Adjacent sequences: A126318 A126319 A126320 this_sequence A126322 A126323 A126324
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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 25 2006
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