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

%I A215981 #16 Jun 16 2018 12:51:19
%S A215981 1,1,2,3,4,7,12,24,48,107,236,552,1302,3160,7742,19321,48630,123868,
%T A215981 317956,823066,2144506,5623757,14828075,39299898,104636891,279793451,
%U A215981 751065461,2023443033,5469566586,14830871803,40330829031,109972410222,300628862481
%N A215981 Number of simple unlabeled graphs on n nodes with exactly 1 connected component that is a tree or a cycle.
%H A215981 Alois P. Heinz, <a href="/A215981/b215981.txt">Table of n, a(n) for n = 1..700</a>
%F A215981 a(1) = a(2) = 1, a(n) = 1 + A000055(n) for n>=3.
%e A215981 a(5) = 4: .o-o-o.  .o-o-o.  .o-o-o.  .o-o-o.
%e A215981           .|  / .  .|    .  .| |  .  . /|  .
%e A215981           .o-o  .  .o-o  .  .o o  .  .o o  .
%p A215981 with(numtheory):
%p A215981 b:= proc(n) option remember; local d, j; `if`(n<=1, n,
%p A215981       (add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1))/(n-1))
%p A215981     end:
%p A215981 a:= proc(n) option remember; local k; `if`(n>2, 1, 0)+ b(n)-
%p A215981       (add(b(k)*b(n-k), k=0..n) -`if`(irem(n, 2)=0, b(n/2), 0))/2
%p A215981     end:
%p A215981 seq(a(n), n=1..40);
%t A215981 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)];
%t A215981 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;
%t A215981 Array[a, 40] (* _Jean-François Alcover_, Mar 21 2017, translated from Maple *)
%Y A215981 Column k=1 of A215977.
%Y A215981 The labeled version is A215851.
%Y A215981 Cf. A000055, A215978.
%K A215981 nonn
%O A215981 1,3
%A A215981 _Alois P. Heinz_, Aug 29 2012