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Search: id:A144258
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| A144258 |
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Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k)=number of forests of trees on n or fewer nodes using a subset of labels 1..n and k edges. |
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+0
2
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| 1, 2, 0, 4, 1, 0, 8, 6, 3, 0, 16, 24, 27, 16, 0, 32, 80, 150, 190, 125, 0, 64, 240, 660, 1335, 1830, 1296, 0, 128, 672, 2520, 7210, 15435, 22449, 16807, 0, 256, 1792, 8736, 33040, 98105, 219912, 335160, 262144, 0, 512, 4608, 28224, 135072, 521010, 1600452
(list; table; graph; listen)
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OFFSET
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0,2
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LINKS
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Index entries for sequences related to trees
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FORMULA
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T(n,0) = 2^n, T(n,k) = 0 if k<0 or n<=k, else T(n,k) = n^(n-2) if k=n-1, else T(n,k) = Sum_{j=0..k} C(n-1,j) T(j+1,j) T(n-1-j,k-j).
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EXAMPLE
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T(3,1) = 6, because there are 6 forests of trees on 3 or less nodes using a subset of labels 1,2,3 and 1 edge:
.1-2. .1... ...2. .1-2. .1.2. .1.2.
..... .|... ../.. ..... .|... ../..
..... .3... .3... .3... .3... .3...
Triangle begins:
1
2, 0
4, 1, 0
8, 6, 3, 0
16, 24, 27, 16, 0
32, 80, 150, 190, 125, 0
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MAPLE
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T:= proc(n, k) option remember; if k=0 then 2^n elif k<0 or n<=k then 0 elif k=n-1 then n^(n-2) else add (binomial (n-1, j) * T(j+1, j) *T(n-1-j, k-j), j=0..k) fi end: seq (seq (T(n, k), k=0..n), n=0..11);
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CROSSREFS
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Columns k=0, 1 give: A000079, A001788. First lower diagonal gives A000272(k+1) with initial term 2. Row sums give: A144259. Cf. A007318, A000142.
Sequence in context: A153345 A140648 A153342 this_sequence A056859 A090888 A154794
Adjacent sequences: A144255 A144256 A144257 this_sequence A144259 A144260 A144261
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KEYWORD
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nonn,tabl
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AUTHOR
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Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 16 2008
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