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A274935
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Number of n-node unlabeled forests with two connected components.
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4
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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
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OFFSET
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0,3
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COMMENTS
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LINKS
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FORMULA
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G.f.: [A(x)^2 + A(x^2)]/2 where A(x) is the o.g.f. for A000055.
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MAPLE
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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:
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MATHEMATICA
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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]]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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