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

1, 3, 7, 25, 127, 777, 5547, 45216, 414144, 4209480, 47009880, 572101920, 7535302560, 106791531840, 1620314539200, 26205248563200, 450022496716800, 8178211565798400, 156798308067609600, 3162998405887488000, 66967168288624128000, 1484773164338365440000

Offset: 2

Author

Alois P. Heinz, Aug 23 2012

Keywords

• nonn

Examples

			a(4) = 7:  .1.2.  .1-2.  .1.2.  o1.2.  .1.2o  .1-2.  .1.2.
.          .|.|.  .....  ..X..  ../|.  .|\..  ..\|.  .|\..
.          .3.4.  .3-4.  .3.4.  .3-4.  .3-4.  o3.4.  .3-4o
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..175

Crossrefs

Column k=2 of A215771.

Programs

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

Maple

a:= proc(n) option remember; `if`(n<7, [0, 0, 1, 3, 7, 25, 127][n+1],
            ((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)))
    end:
seq(a(n), n=2..30);

Mathematica

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 *)

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

Formula

See Maple program.

a(n) ~ (n-1)! * (log(n) + 3/2 + gamma)/4, where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Apr 27 2015