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A274937 Number of unlabeled forests on n nodes that have exactly two non-empty components. 5
0, 0, 1, 1, 2, 3, 6, 11, 23, 46, 99, 216, 488, 1121, 2644, 6334, 15437, 38132, 95368, 241029, 614968, 1582030, 4100157, 10697038, 28075303, 74086468, 196470902, 523383136, 1400051585, 3759508536, 10131097618, 27391132238, 74283552343, 202030012554, 550934060120, 1506161266348 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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(n) = A095133(n,2). - R. J. Mathar, Jul 20 2016

MAPLE

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, 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], 0, Function[t, t*(t-1)/2][g[n/2]]];

Table[a[n], {n, 0, 40}] (* Jean-Fran├žois Alcover, Mar 14 2017, after Alois P. Heinz *)

CROSSREFS

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

Sequence in context: A036592 A036656 A001190 * A199142 A090344 A277795

Adjacent sequences:  A274934 A274935 A274936 * A274938 A274939 A274940

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jul 19 2016

STATUS

approved

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Last modified September 24 04:27 EDT 2017. Contains 292403 sequences.