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A215862 Number of simple labeled graphs on n+2 nodes with exactly n connected components that are trees or cycles.

%I A215862 #31 Oct 30 2024 08:05:15
%S A215862 0,4,19,55,125,245,434,714,1110,1650,2365,3289,4459,5915,7700,9860,
%T A215862 12444,15504,19095,23275,28105,33649,39974,47150,55250,64350,74529,
%U A215862 85869,98455,112375,127720,144584,163064,183260,205275,229215,255189,283309,313690,346450
%N A215862 Number of simple labeled graphs on n+2 nodes with exactly n connected components that are trees or cycles.
%C A215862 Partial sums of A077414. - _Bruno Berselli_, Jul 30 2015
%H A215862 Alois P. Heinz, <a href="/A215862/b215862.txt">Table of n, a(n) for n = 0..1000</a>
%H A215862 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A215862 G.f.: (x-4)*x/(x-1)^5.
%F A215862 a(n) = C(n+2,3)*(3*n+13)/4.
%F A215862 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 A215862 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 A215862 a(n) = 4*A000332(n+2) - A000332(n+1). - _R. J. Mathar_, Aug 12 2013
%F A215862 a(n) = Sum_{i=0..n} (3+i)*A000217(i). - _Bruno Berselli_, Apr 29 2014
%e A215862 a(1) = 4:
%e A215862 .1-2.  .1-2.  .1-2.  .1 2.
%e A215862 .|/ .  .|. .  . / .  .|/ .
%e A215862 .3...  .3...  .3...  .3...
%p A215862 a:= n-> binomial(n+2,3)*(3*n+13)/4:
%p A215862 seq(a(n), n=0..40);
%t A215862 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 A215862 A diagonal of A215861.
%Y A215862 Regarding the sixth formula, see similar sequences listed in A241765.
%Y A215862 Cf. A000332, A077414.
%K A215862 nonn,easy
%O A215862 0,2
%A A215862 _Alois P. Heinz_, Aug 25 2012