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Search: id:A137916
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| A137916 |
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Number of labeled graphs on [n] with unicyclic components. |
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+0
3
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| 0, 0, 1, 15, 222, 3670, 68820, 1456875, 34506640, 906073524, 26154657270, 823808845585, 28129686128940, 1035350305641990, 40871383866109888, 1722832666898627865, 77242791668604946560, 3670690919234354407000
(list; graph; listen)
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OFFSET
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1,4
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COMMENT
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The first values are row sums of A106239.
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LINKS
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Wikipedia, Pseudo forest
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FORMULA
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a(n)= Sum N/D over the partitions of n: 1p_1+2p_2+ ... +np_n, with parts >=3, where N = n!*product_{1=<i<=n}= A057500(i)^p_i and D = product_{1=<i<=n}(p_i!(i!)^p_i).
a(n) = A144228(n,n). [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 15 2008]
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EXAMPLE
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E.g. a(6) = 3670 because there are 3660 distinct labeled unicycles with 6 vertices and only 10 ways to label two triangles.
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MAPLE
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cy:= proc(n) option remember; local t; binomial(n-1, 2) *add ((n-3)! /(n-2-t)! *n^(n-2-t), t=1..n-2) end: T:= proc(n, k) option remember; local j; if k=0 then 1 elif k<0 or n<k then 0 else add (binomial (n-1, j) *((j+1)^(j-1) *T(n-j-1, k-j) +cy(j+1) *T(n-j-1, k-j-1)), j=0..k) fi end: a:= n-> T(n, n): seq (a(n), n=1..20); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 15 2008]
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CROSSREFS
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Cf. A057500, A106239.
Diagonal of A144228. [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 15 2008]
Sequence in context: A027843 A027840 A057500 this_sequence A171320 A078364 A012852
Adjacent sequences: A137913 A137914 A137915 this_sequence A137917 A137918 A137919
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KEYWORD
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easy,nonn
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AUTHOR
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Washington G. Bomfim (webonfim(AT)bol.com.br), Feb 22 2008
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