A215981 Number of simple unlabeled graphs on n nodes with exactly 1 connected component that is a tree or a cycle.

1, 1, 2, 3, 4, 7, 12, 24, 48, 107, 236, 552, 1302, 3160, 7742, 19321, 48630, 123868, 317956, 823066, 2144506, 5623757, 14828075, 39299898, 104636891, 279793451, 751065461, 2023443033, 5469566586, 14830871803, 40330829031, 109972410222, 300628862481

Offset: 1

Author

Alois P. Heinz, Aug 29 2012

Keywords

• nonn

Examples

			a(5) = 4: .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 = 1..700

Crossrefs

Column k=1 of A215977.

The labeled version is A215851.

Cf. A000055, A215978.

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:
a:= 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:
seq(a(n), n=1..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)];
a[n_] := a[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;
Array[a, 40] (* Jean-François Alcover, Mar 21 2017, translated from Maple *)

Formula

a(1) = a(2) = 1, a(n) = 1 + A000055(n) for n>=3.