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A120986
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Triangle read by rows: T(n,k) is the number of ternary trees with n edges and having k middle edges (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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1
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1, 2, 1, 5, 6, 1, 14, 28, 12, 1, 42, 120, 90, 20, 1, 132, 495, 550, 220, 30, 1, 429, 2002, 3003, 1820, 455, 42, 1, 1430, 8008, 15288, 12740, 4900, 840, 56, 1, 4862, 31824, 74256, 79968, 42840, 11424, 1428, 72, 1, 16796, 125970, 348840, 465120, 325584
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OFFSET
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0,2
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COMMENTS
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Row sums yield A001764. T(n,0)=A000108(n+1) (the Catalan numbers). T(n,k)=A108767(n+1,n+1-k) Sum(k*T(n,k),k>=1)=binom(3n+2,n-1)=A013698(n).
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REFERENCES
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Chao-Jen Wang, Applications of the Goulden-Jackson cluster method to counting Dyck paths by occurrences of subwords, http://people.brandeis.edu/~gessel/homepage/students/wangthesis.pdf.
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LINKS
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Table of n, a(n) for n=0..49.
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FORMULA
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T(n,k)=(1/(n+1))*binomial(n+1,k)*binomial(2(n+1),n-k). G.f.=G=G(t,z) satisfies G=(1+tzG)(1+zG)^2.
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EXAMPLE
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Triangle starts:
1;
2,1;
5,6,1;
14,28,12,1;
42,120,90,20,1;
132,495,550,220,30,1;
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MAPLE
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T:=(n, k)->binomial(n+1, k)*binomial(2*(n+1), n-k)/(n+1): 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
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Cf. A001764, A000108, A108767, A013698.
Sequence in context: A179457 A107783 A047887 * A095801 A128567 A217204
Adjacent sequences: A120983 A120984 A120985 * A120987 A120988 A120989
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch, Jul 21 2006
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STATUS
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approved
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