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

2,2

Links

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

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

Example

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

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 *)

Prog

(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
(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

Crossrefs

Column k=2 of A215771.

Sequence in context: A096648 A156100 A245115 * A019056 A065163 A292925

Adjacent sequences: A215769 A215770 A215771 * A215773 A215774 A215775

Keyword

nonn

Author

Alois P. Heinz, Aug 23 2012

Status

approved