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A137916 Number of labeled graphs on [n] with unicyclic components. 6
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; text; internal format)
OFFSET

1,4

COMMENTS

The first values are row sums of A106239.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..150

Wikipedia, Pseudo forest

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 (labeled rooted trees). - Geoffrey Critzer, Jan 24 2012

a(n) ~ 2^(-1/4)*exp(-3/4)*GAMMA(3/4)*n^(n-1/4)/sqrt(Pi) * (1-7*Pi/(12*GAMMA(3/4)^2*sqrt(n))). - Vaclav Kotesovec, Aug 16 2013

EXAMPLE

a(6) = 3670 because there are 3660 distinct labeled unicycles with 6 vertices and only 10 ways to label two triangles.

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

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 *)

CROSSREFS

Cf. A057500, A106239.

Diagonal of A144228. - Alois P. Heinz, Sep 15 2008

Sequence in context: A027840 A279530 A057500 * A218696 A297669 A171320

Adjacent sequences:  A137913 A137914 A137915 * A137917 A137918 A137919

KEYWORD

easy,nonn

AUTHOR

Washington Bomfim, Feb 22 2008

STATUS

approved

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Last modified October 22 06:54 EDT 2019. Contains 328315 sequences. (Running on oeis4.)