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A274935 Number of n-node unlabeled forests with two connected components. 4
1, 1, 2, 2, 4, 6, 12, 22, 46, 93, 205, 451, 1039, 2422, 5803, 14075, 34757, 86761, 219235, 558984, 1438033, 3726535, 9723913, 25525112, 67375200, 178723358, 476264352, 1274448596, 3423494617, 9229075121, 24961969420, 67721961268, 184255962564, 502658875034, 1374713691841, 3768527610094, 10353602341313 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

One of the components may be empty (the null graph): a(n) = A000055(n) + A274937(n). - R. J. Mathar, Aug 15 2017

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.

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-Fran├žois Alcover, Mar 15 2017, after Alois P. Heinz *)

CROSSREFS

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

Sequence in context: A238014 A052953 A128209 * A188538 A282164 A268175

Adjacent sequences:  A274932 A274933 A274934 * A274936 A274937 A274938

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jul 19 2016

STATUS

approved

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Last modified October 16 02:52 EDT 2019. Contains 328038 sequences. (Running on oeis4.)