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Search: id:A126179
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| A126179 |
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Triangle read by rows: T(n,k) is number of hex trees with n edges and k branches (1<=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 median 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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2
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| 3, 9, 1, 27, 6, 3, 81, 27, 27, 2, 243, 108, 162, 24, 6, 729, 405, 810, 180, 90, 5, 2187, 1458, 3645, 1080, 810, 90, 15, 6561, 5103, 15309, 5670, 5670, 945, 315, 14, 19683, 17496, 61236, 27216, 34020, 7560, 3780, 336, 42, 59049, 59049, 236196, 122472
(list; table; graph; listen)
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OFFSET
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1,1
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COMMENT
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Sum of terms in row n = A002212(n+1). T(n,1)=3^n (A000244). T(2n,2n)=c(n); T(2n+1,2n+1)=3*c(n), where c(n)=binom(2n,n)/(n+1) is a Catalan number (A000108). Sum(k*T(n,k),k=1..n)=A126180(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.
J. Riordan, Enumeration of plane trees by branches and endpoints, J. Comb. Theory (A) 19, 1975, 214-222.
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FORMULA
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T(n,k)=3^(n-k+1)*binomial(n-1,k-1)*c((k-1)/2) if k is odd; T(n,k)=3^(n-k)*binomial(n-1,k-1)*c(k/2) if k is even; c(m)=binom(2m,m)/(m+1) is a Catalan number. G.f.=[(1-3z+3tz)/(1-3z)]C(t^2*z^2/(1-3z)^2)-1, where C(z)=[1-sqrt(1-4z)]/(2z) is the Catalan function. G.f.=(1-3z+3tz)[1-3z-sqrt((1-3z)^2-4t^2*z^2)]/(2t^2*z^2)-1;
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EXAMPLE
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Triangle starts:
3;
9,1;
27,6,3;
81,27,27,2;
243,108,162,24,6;
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MAPLE
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c:=n->binomial(2*n, n)/(n+1): T:=proc(n, k) if k mod 2 = 0 then 3^(n-k)*binomial(n-1, k-1)*c(k/2) else 3^(n-k+1)*binomial(n-1, k-1)*c((k-1)/2) fi end: for n from 1 to 11 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A002212, A000244, A000108, A126180.
Sequence in context: A119796 A019817 A080322 this_sequence A128727 A126177 A128733
Adjacent sequences: A126176 A126177 A126178 this_sequence A126180 A126181 A126182
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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 19 2006
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