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%I #27 Aug 01 2015 10:36:08
%S 0,4,19,55,125,245,434,714,1110,1650,2365,3289,4459,5915,7700,9860,
%T 12444,15504,19095,23275,28105,33649,39974,47150,55250,64350,74529,
%U 85869,98455,112375,127720,144584,163064,183260,205275,229215,255189,283309,313690,346450
%N Number of simple labeled graphs on n+2 nodes with exactly n connected components that are trees or cycles.
%C Partial sums of A077414. [_Bruno Berselli_, Jul 30 2015]
%H Alois P. Heinz, <a href="/A215862/b215862.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5, -10, 10, -5, 1).
%F G.f.: (x-4)*x/(x-1)^5.
%F a(n) = C(n+2,3)*(3*n+13)/4.
%F a(n) = 5*a(n-1)- 10*a(n-2)+ 10*a(n-3) -5*a(n-4)+a(n-5), n>4. - _Harvey P. Dale_, Sep 10 2012
%F a(n) = 1/n! * Sum_{j=0..n} C(n,j)*(-1)^(n-j)*(j)^(n+1)*(j-1)). - _Vladimir Kruchinin_, Jun 06 2013
%F a(n) = 4*A000332(n+2) - A000332(n+1). - _R. J. Mathar_, Aug 12 2013
%F a(n) = Sum_{i=0..n} (3+i)*A000217(i). [_Bruno Berselli_, Apr 29 2014]
%e a(1) = 4:
%e .1-2. .1-2. .1-2. .1 2.
%e .|/ . .|. . . / . .|/ .
%e .3... .3... .3... .3...
%p a:= n-> binomial(n+2,3)*(3*n+13)/4:
%p seq(a(n), n=0..40);
%t Table[Binomial[n+2,3] (3n+13)/4,{n,0,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{0,4,19,55,125},40] (* _Harvey P. Dale_, Sep 10 2012 *)
%Y A diagonal of A215861.
%Y Regarding the sixth formula, see similar sequences listed in A241765.
%Y Cf. A000332, A077414.
%K nonn,easy
%O 0,2
%A _Alois P. Heinz_, Aug 25 2012
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