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

0,4

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.

a(2n+1) = A274935(2n+1). a(2n) = A274935(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=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 *)

Crossrefs

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

Sequence in context: A132831 A354208 A007477 * A244521 A096202 A036653

Adjacent sequences: A274933 A274934 A274935 * A274937 A274938 A274939

Keyword

nonn

Author

N. J. A. Sloane, Jul 19 2016

Status

approved