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A215772 Number of undirected labeled graphs on n nodes with exactly 2 cycle graphs as connected components.

%I A215772 #12 Sep 08 2022 08:46:03
%S A215772 1,3,7,25,127,777,5547,45216,414144,4209480,47009880,572101920,
%T A215772 7535302560,106791531840,1620314539200,26205248563200,450022496716800,
%U A215772 8178211565798400,156798308067609600,3162998405887488000,66967168288624128000,1484773164338365440000
%N A215772 Number of undirected labeled graphs on n nodes with exactly 2 cycle graphs as connected components.
%H A215772 Alois P. Heinz, <a href="/A215772/b215772.txt">Table of n, a(n) for n = 2..175</a>
%F A215772 See Maple program.
%F A215772 a(n) ~ (n-1)! * (log(n) + 3/2 + gamma)/4, where gamma is the Euler-Mascheroni constant (A001620). - _Vaclav Kotesovec_, Apr 27 2015
%e A215772 a(4) = 7:  .1.2.  .1-2.  .1.2.  o1.2.  .1.2o  .1-2.  .1.2.
%e A215772 .          .|.|.  .....  ..X..  ../|.  .|\..  ..\|.  .|\..
%e A215772 .          .3.4.  .3-4.  .3.4.  .3-4.  .3-4.  o3.4.  .3-4o
%p A215772 a:= proc(n) option remember; `if`(n<7, [0, 0, 1, 3, 7, 25, 127][n+1],
%p A215772             ((2*n^3-21*n^2+65*n-58)*a(n-1)
%p A215772       -(n^4-13*n^3+60*n^2-116*n+80)*a(n-2))/((n-3)*(n-6)))
%p A215772     end:
%p A215772 seq(a(n), n=2..30);
%t A215772 Join[{1, 3, 7, 25, 127}, RecurrenceTable[{a[n] == ((2*n^3 - 21*n^2 + 65*n - 58)*a[n-1] - (n^4 - 13*n^3 + 60*n^2 - 116*n + 80)*a[n-2])/((n-3)*(n- 6)), a[7] == 777, a[8] == 5547}, a, {n, 7, 20}]] (* _G. C. Greubel_, Aug 30 2018 *)
%o A215772 (PARI) m=30; v=concat([1,3,7,25,127, 777, 5547], vector(m-6)); for(n=7, m, v[n] = ((2*n^3-21*n^2+65*n-58)*v[n-1]-(n^4-13*n^3+60*n^2-116*n +80)*v[n-2] )/((n-3)*(n-6))); v \\ _G. C. Greubel_, Aug 30 2018
%o A215772 (Magma) I:=[1, 3, 7, 25, 127, 777,5547]; [n le 7 select I[n] else ((2*n^3 - 21*n^2 + 65*n - 58)*Self(n-1) - (n^4 - 13*n^3 + 60*n^2 - 116*n + 80)*Self(n-2))/((n-3)*(n- 6)): n in [1..30]]; // _G. C. Greubel_, Aug 30 2018
%Y A215772 Column k=2 of A215771.
%K A215772 nonn
%O A215772 2,2
%A A215772 _Alois P. Heinz_, Aug 23 2012