A274938 Number of unlabeled forests with n nodes that have two components, neither of which is the empty graph.

0, 0, 0, 1, 1, 3, 5, 11, 21, 46, 96, 216, 482, 1121, 2633, 6334, 15414, 38132, 95321, 241029, 614862, 1582030, 4099922, 10697038, 28074752, 74086468, 196469601, 523383136, 1400048426, 3759508536, 10131089877, 27391132238, 74283533023, 202030012554, 550934011491, 1506161266348

Offset

0,6

Links

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

Formula

G.f.: [A(x)^2 - A(x^2)]/2 where A(x) is the o.g.f. for A000055 without the initial constant 1.

a(2n+1) = A274937(2n+1). a(2n) = A274937(2n)-A000055(n). - R. J. Mathar, Jul 20 2016

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=1..n/2)-
      `if`(n::odd or n=0, 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, 1, n/2}]-If[OddQ[n] || n==0, 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 *)

Crossrefs

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

Sequence in context: A283905 A146574 A007873 * A283603 A283818 A352006

Adjacent sequences: A274935 A274936 A274937 * A274939 A274940 A274941

Keyword

nonn

Author

N. J. A. Sloane, Jul 19 2016

Status

approved