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A121685
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Triangle read by rows: T(n,k) is the number of binary trees having n edges and k branches (1<=k<=n). A binary tree is a rooted tree in which each vertex has at most two children and each child of a vertex is designated as its left or right child.
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1
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2, 4, 1, 8, 4, 2, 16, 12, 12, 2, 32, 32, 48, 16, 4, 64, 80, 160, 80, 40, 5, 128, 192, 480, 320, 240, 60, 10, 256, 448, 1344, 1120, 1120, 420, 140, 14, 512, 1024, 3584, 3584, 4480, 2240, 1120, 224, 28, 1024, 2304, 9216, 10752, 16128, 10080, 6720, 2016, 504, 42
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| The row sums are the Catalan numbers (A000108). T(n,1)=2^n = A000079(n). T(n,n)=A089408(n+1). Sum(k*T(n,k),k=1..n)=A121686(n).
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FORMULA
| T(n,k)=2^(n-k)*c(k/2)*binomial(n-1,k-1) if k is even and 2^(n-k+1)*c((k-1)/2)*binomial(n-1,k-1) if k is odd, where c(m)=binomial(2m,m)/(m+1) are the Catalan numbers (A000108). G.f.=(1-2z+2tz)(1-2z-sqrt[(1-2z)^2-4t^2*z^2])/(2t^2*z^2) - 1.
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EXAMPLE
| Triangle starts:
2;
4,1;
8,4,2;
16,12,12,2;
32,32,48,16,4;
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MAPLE
| c:=n->binomial(2*n, n)/(n+1): T:=proc(n, k) if k mod 2 = 0 then c(k/2)*binomial(n-1, k-1)*2^(n-k) else c((k-1)/2)*binomial(n-1, k-1)*2^(n-k+1) 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
| Cf. A000108, A000079, A089408, A121686.
Sequence in context: A173122 A065278 A112931 * A125810 A152195 A133156
Adjacent sequences: A121682 A121683 A121684 * A121686 A121687 A121688
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 15 2006
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