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A215853 Number of simple labeled graphs on n nodes with exactly 3 connected components that are trees or cycles.

%I A215853 #15 Apr 01 2017 19:01:17
%S A215853 1,6,55,540,6412,90734,1515097,29368155,649910349,16178495157,
%T A215853 447436384356,13607804913248,451277483034618,16204761730619392,
%U A215853 626327433705523558,25924177756443661632,1144012780063556028591,53615833082093775740400,2659498185704802765924159
%N A215853 Number of simple labeled graphs on n nodes with exactly 3 connected components that are trees or cycles.
%H A215853 Alois P. Heinz, <a href="/A215853/b215853.txt">Table of n, a(n) for n = 3..145</a>
%F A215853 a(n) ~ c * n^(n-2), where c = 0.130848879059... . - _Vaclav Kotesovec_, Sep 07 2014
%e A215853 a(4) = 6:
%e A215853 .1-2.  .1 2.  .1 2.  .1 2.  .1 2.  .1 2.
%e A215853 .   .  .  |.  .   .  .|  .  . \ .  . / .
%e A215853 .4 3.  .4 3.  .4-3.  .4 3.  .4 3.  .4 3.
%p A215853 T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
%p A215853       `if`(n=0, 1, add(binomial(n-1, i)*T(n-1-i, k-1)*
%p A215853       `if`(i<2, 1, i!/2 +(i+1)^(i-1)), i=0..n-k)))
%p A215853     end:
%p A215853 a:= n-> T(n, 3):
%p A215853 seq(a(n), n=3..25);
%t A215853 T[n_, k_] := T[n, k] = If[k<0 || k>n, 0, If[n == 0, 1, Sum[Binomial[n-1, i]*T[n-1-i, k-1]*If[i<2, 1, i!/2 + (i+1)^(i-1)], {i, 0, n-k}]]];
%t A215853 a[n_] := T[n, 3];
%t A215853 Table[a[n], {n, 3, 25}] (* _Jean-François Alcover_, Apr 01 2017, translated from Maple *)
%Y A215853 Column k=3 of A215861.
%Y A215853 The unlabeled version is A215983.
%K A215853 nonn
%O A215853 3,2
%A A215853 _Alois P. Heinz_, Aug 25 2012