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%I #21 Aug 12 2017 05:04:10
%S 1,1,1,2,8,38,208,1348,10126,86174,819134,8604404,98981944,1237575268,
%T 16710431992,242337783032,3756693451772,61991635990652,
%U 1084943597643964,20072853005524696,391443701509660096,8024999955144721256,172544980412641191776
%N Number of labeled 2-regular graphs with no multiple edges, but loops are allowed (i.e., each vertex is endpoint of two (usual) edges or one loop).
%H Alois P. Heinz, <a href="/A108246/b108246.txt">Table of n, a(n) for n = 0..200</a>
%F Linear recurrence satisfied by a(n): {a(2) = 1, a(0) = 1, (-n^2 - 3*n - 2)*a(n) + (4 + 2*n)*a(n+1) + (-2*n-6)*a(n+2) + 2*a(n+3), a(1) = 1}.
%F E.g.f.: exp(-t^2/4 + t/2)/sqrt(1-t). - _Vladeta Jovovic_, Aug 14 2006
%F a(n) ~ sqrt(2)*n^n/exp(n-1/4). - _Vaclav Kotesovec_, Oct 17 2012
%e a(3) = 2: {(1,2) (2,3) (1,3)}, {(1,1) (2,2) (3,3)}.
%p b:= proc(n) option remember; if n=0 then 1 elif n<3 then 0 else (n-1) *(b(n-1) +b(n-3) *(n-2)/2) fi end: a:= proc(n) add(b(k) *binomial(n,k), k=0..n) end: seq(a(n), n=0..30); # _Alois P. Heinz_, Sep 12 2008
%t CoefficientList[Series[E^(-x^2/4+x/2)/Sqrt[1-x], {x, 0, 20}], x]* Table[n!, {n, 0, 20}] (* _Vaclav Kotesovec_, Oct 17 2012 *)
%Y Cf. A000985, A002137.
%Y Binomial transform of A001205.
%Y Row sums of A144161. - _Alois P. Heinz_, Jun 01 2009
%K nonn
%O 0,4
%A _Marni Mishna_, Jun 17 2005
%E More terms from _Alois P. Heinz_, Sep 12 2008
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