A143900 - OEIS

%I #18 Mar 02 2019 02:19:46

%S 0,0,0,1,26,733,29836,2060191,267873508,68709450231,35184166480296,

%T 36028792251523289,73786976171465003256,302231454900131663566437,

%U 2475880078570650265515241808,40564819207303337099536803011071,1329227995784915872766249150185503728

%N Number of simple graphs on n labeled nodes containing at least one cycle subgraph, also row sums of A143899.

%H Robert G. Wilson v, <a href="/A143900/b143900.txt">Table of n, a(n) for n = 0..82</a> (first 51 terms from Alois P. Heinz)

%F a(n) = A006125(n) - A001858(n).

%F a(n) = Sum_{k=3..C(n,2)} A143899(n,k).

%e a(3) = 1, because 1 simple graph on 3 nodes with 3 edges contains a cycle subgraph:

%e ..1-2..

%e ..|/...

%e ..3....

%p graphs:= n-> 2^binomial(n,2): forests:= n-> add(add(binomial(m,j) *binomial(n-1, n-m-j) *n^(n-m-j) *(m+j)!/ (-2)^j/ m!, j=0..m), m=0..n): a:= n-> graphs(n) -forests(n): seq(a(n), n=0..18);

%t graphs[n_] := 2^Binomial[n, 2]; forests[n_] := Sum[Binomial[m, j]* Binomial[n-1, n-m-j]*n^(n-m-j)*(m+j)!/(-2)^j/m!, {m, 0, n}, {j, 0, m}]; a[0] = 0; a[n_] := graphs[n] - forests[n]; Table[a[n], {n, 0, 18}] (* _Jean-François Alcover_, Feb 25 2017, after _Alois P. Heinz_ *)

%Y Row sums of A143899.

%Y Cf. A006125, A001858, A007318.

%K nonn

%O 0,5

%A _Alois P. Heinz_, Sep 04 2008