A274936 Number of n-node unlabeled forests that have 2 non-isomorphic components.

0, 1, 1, 2, 3, 6, 11, 22, 44, 93, 202, 451, 1033, 2422, 5792, 14075, 34734, 86761, 219188, 558984, 1437927, 3726535, 9723678, 25525112, 67374649, 178723358, 476263051, 1274448596, 3423491458, 9229075121, 24961961679, 67721961268, 184255943244, 502658875034, 1374713643212

Offset: 0

Author

N. J. A. Sloane, Jul 19 2016

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000055, A274935-A274938. [A274935, A274936, A274937, A274938] are analogs for forests of [A275165, A275166, A216785, A274934] for graphs.

Programs

Maple

with(numtheory):
b:= proc(n) option remember; `if`(n<2, n, (add(add(d*
      b(d), d=divisors(j))*b(n-j), j=1..n-1))/(n-1))
    end:
g:= proc(n) option remember; `if`(n=0, 1, b(n)-add(b(j)*
      b(n-j), j=0..n/2)+`if`(n::odd, 0, (t->t*(t+1)/2)(b(n/2))))
    end:
a:= proc(n) option remember; add(g(j)*g(n-j), j=0..n/2)-
      `if`(n::odd, 0, (t-> t*(t+1)/2)(g(n/2)))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 20 2016

Mathematica

b[n_] := b[n] = If[n<2, n, Sum[DivisorSum[j, #*b[#]&]*b[n-j], {j, 1, n-1}]/(n-1)];
g[n_] := g[n] = If[n==0, 1, b[n]-Sum[b[j]*b[n-j], {j, 0, n/2}]+If[OddQ[n], 0, Function[t, t*(t+1)/2][b[n/2]]]];
a[n_] := a[n] = Sum[g[j]*g[n-j], {j, 0, n/2}]-If[OddQ[n], 0, Function[t, t*(t+1)/2][g[n/2]]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 15 2017, after Alois P. Heinz *)

Formula

G.f.: [A(x)^2 - A(x^2)]/2 where A(x) is the o.g.f. for A000055.

a(2n+1) = A274935(2n+1). a(2n) = A274935(2n)-A000055(n). - R. J. Mathar, Jul 20 2016