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A137916
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Number of labeled graphs on [n] with unicyclic components.
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3
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0, 0, 1, 15, 222, 3670, 68820, 1456875, 34506640, 906073524, 26154657270, 823808845585, 28129686128940, 1035350305641990, 40871383866109888, 1722832666898627865, 77242791668604946560, 3670690919234354407000, 184312149879830557190940, 9751080154504005703189791
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| The first values are row sums of A106239.
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LINKS
| Alois P. Heinz, Table of n, a(n) for n = 1..150
Wikipedia, Pseudo forest
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FORMULA
| 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). - Alois P. Heinz, Sep 15 2008
E.g.f.: exp(B(T(x))) where B(x)= (log(1/(1-x))-x-x^2/2)/2 and T(x) is the e.g.f. for A000169 (labelled rooted trees). - Geoffrey Critzer, Jan 24 2012
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EXAMPLE
| 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
| 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); # Alois P. Heinz, Sep 15 2008
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MATHEMATICA
| nn = 20; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}]; Drop[Range[0, nn]! CoefficientList[Series[Exp[Log[1/(1 - t)]/2 - t/2 - t^2/4], {x, 0, nn}], x], 1] (* Geoffrey Critzer, Jan 24 2012 *)
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CROSSREFS
| Cf. A057500, A106239.
Diagonal of A144228. - Alois P. Heinz, Sep 15 2008
Sequence in context: A027843 A027840 A057500 * A171320 A078364 A012852
Adjacent sequences: A137913 A137914 A137915 * A137917 A137918 A137919
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KEYWORD
| easy,nonn
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AUTHOR
| Washington G. Bomfim (webonfim(AT)bol.com.br), Feb 22 2008
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