A215982 Number of simple unlabeled graphs on n nodes with exactly 2 connected components that are trees or cycles.

1, 1, 3, 5, 10, 17, 33, 62, 127, 267, 587, 1326, 3085, 7326, 17731, 43585, 108563, 273544, 696113, 1787042, 4623125, 12043071, 31565842, 83200763, 220413272, 586625403, 1567930743, 4207181144, 11329835687, 30613313339, 82975300030, 225552632043, 614787508640

Offset: 2

Author

Alois P. Heinz, Aug 29 2012

Keywords

• nonn

Examples

			a(5) = 5: .o-o o.  .o-o o.  .o-o o.  .o o-o.  .o o-o.
          .| |  .  .|    .  .|\   .  .|\   .  .|    .
          .o-o  .  .o-o  .  .o o  .  .o-o  .  .o-o  .
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..650

Crossrefs

Column k=2 of A215977.

The labeled version is A215852.

Programs

Maple

with(numtheory):
b:= proc(n) option remember; local d, j; `if` (n<=1, n,
      (add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1))/(n-1))
    end:
g:= proc(n) option remember; local k; `if`(n>2, 1, 0)+ b(n)-
      (add(b(k)*b(n-k), k=0..n) -`if`(irem(n, 2)=0, b(n/2), 0))/2
    end:
p:= proc(n, i, t) option remember; `if`(n p(n, n, 2):
seq(a(n), n=2..40);

Mathematica

b[n_] := b[n] = If[n <= 1, n, Sum[Sum[d*b[d], {d, Divisors[j]}]*b[n-j], {j, 1, n-1}]/(n-1)]; g[n_] := g[n] = If[n>2, 1, 0]+b[n]-(Sum [b[k]*b[n-k], {k, 0, n}] - If[Mod[n, 2] == 0, b[n/2], 0])/2; p[n_, i_, t_] := p[n, i, t] = If[nJean-François Alcover, Dec 04 2014, translated from Maple *)

Python

from sympy.core.cache import cacheit
from sympy import binomial, divisors
@cacheit
def b(n): return n if n<2 else sum([sum([d*b(d) for d in divisors(j)])*b(n - j) for j in range(1, n)])//(n - 1)
@cacheit
def g(n): return (1 if n>2 else 0) + b(n) - (sum([b(k)*b(n - k) for k in range(n + 1)]) - (b(n//2) if n%2==0 else 0))//2
@cacheit
def p(n, i, t): return 0 if nIndranil Ghosh, Aug 07 2017

Formula

a(n) ~ c * d^n / n^(5/2), where d = A051491 = 2.9557652856519949747148..., c = 0.3339525664158379... . - Vaclav Kotesovec, Sep 07 2014