A215852 Number of simple labeled graphs on n nodes with exactly 2 connected components that are trees or cycles.

1, 3, 19, 135, 1267, 15029, 218627, 3783582, 75956664, 1734309929, 44357222772, 1255715827483, 38971877812380, 1315634598619830, 47994245894462576, 1881406032047006812, 78870928008704884848, 3520953336130828001295, 166762291211479030734580

Offset

2,2

Links

Alois P. Heinz, Table of n, a(n) for n = 2..145

Formula

a(n) ~ c * n^(n-2), where c = 0.511564031298... . - Vaclav Kotesovec, Sep 07 2014

Example

a(3) = 3:
.1 2.  .1-2.  .1 2.
.|. .  . . .  . / .
.3...  .3...  .3...

Maple

T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      `if`(n=0, 1, add(binomial(n-1, i)*T(n-1-i, k-1)*
      `if`(i<2, 1, i!/2 +(i+1)^(i-1)), i=0..n-k)))
    end:
a:= n-> T(n, 2):
seq(a(n), n=2..25);

Mathematica

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}]]]; Table[T[n, 2], {n, 2, 50}] (* Indranil Ghosh, Aug 07 2017, after Maple *)

Prog

(Python)
from sympy.core.cache import cacheit
from sympy import binomial, factorial as f
@cacheit
def T(n, k): return 0 if k<0 or k>n else 1 if n==0 else sum([binomial(n - 1, i)*T(n - 1 - i, k - 1)*(1 if i<2 else f(i)//2 + (i + 1)**(i - 1)) for i in range(n - k + 1)])
def a(n): return T(n , 2)
print([a(n) for n in range(2, 51)]) # Indranil Ghosh, Aug 07 2017, after maple code

Crossrefs

Column k=2 of A215861.

The unlabeled version is A215982.

Sequence in context: A091346 A305550 A035086 * A105797 A278189 A221297

Adjacent sequences: A215849 A215850 A215851 * A215853 A215854 A215855

Keyword

nonn

Author

Alois P. Heinz, Aug 25 2012

Status

approved