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A215862 Number of simple labeled graphs on n+2 nodes with exactly n connected components that are trees or cycles. 7

%I

%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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Last modified March 30 16:16 EDT 2020. Contains 333127 sequences. (Running on oeis4.)