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A120981
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Triangle read by rows: T(n,k) is the number of ternary trees with n edges and having k vertices of outdegree 1 (n>=0, k>=0). A ternary tree is a rooted tree in which each vertex has at most three children and each child of a vertex is designated as its left or middle or right child.
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4
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1, 0, 3, 3, 0, 9, 1, 27, 0, 27, 18, 12, 162, 0, 81, 15, 270, 90, 810, 0, 243, 138, 270, 2430, 540, 3645, 0, 729, 189, 2898, 2835, 17010, 2835, 15309, 0, 2187, 1218, 4536, 34776, 22680, 102060, 13608, 61236, 0, 6561, 2280, 32886, 61236, 312984, 153090, 551124
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Row sums yield A001764. T(n,0)=A120984(n). Sum(k*T(n,k),k>=1)=3*binom(3n,n-1)=3*A004319(n).
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FORMULA
| T(n,k)=(1/(n+1))*binomial(n+1,k)*sum(3^(2k-n+3j)*binomial(n+1-k,j)*binomial(j,n-k-2j), j=0..n+1-k). G.f.=G=G(t,z) satisfies G = 1+3tzG+3z^2*G^2+z^3*G^3.
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EXAMPLE
| T(2,0)=3 because we have (Q,L,M), (Q,L,R) and (Q,M,R), where Q denotes the root and L (M,R) denotes a left (middle, right) child of Q.
Triangle starts:
1;
0,3;
3,0,9;
1,27,0,27;
18,12,162,0,81;
15,270,90,810,0,243;
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MAPLE
| T:=proc(n, k) if k<=n then (1/(n+1))*binomial(n+1, k)*sum(3^(3*j-n+2*k)*binomial(n+1-k, j)*binomial(j, n-k-2*j), j=0..n+1-k) else 0 fi end: for n from 0 to 10 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form
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CROSSREFS
| Cf. A001764, A004319, A120429, A120982, A120983, A120984.
Sequence in context: A176276 A200701 A169670 * A100543 A039928 A137259
Adjacent sequences: A120978 A120979 A120980 * A120982 A120983 A120984
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 21 2006
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