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A035055
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Number of forests of different trees.
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2
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1, 1, 1, 2, 3, 6, 12, 24, 49, 105, 231, 517, 1188, 2783, 6643, 16101, 39606, 98605, 248287, 631214, 1618878, 4183964, 10889305, 28517954, 75111521, 198851386, 528929895, 1412993746, 3789733399, 10201625514, 27555373561, 74664487653, 202908119046, 552939614498
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OFFSET
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0,4
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LINKS
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FORMULA
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a(n) ~ c * d^n / n^(5/2), where d = A051491 = 2.9557652856519949747148175..., c = 0.89246007934060351292465521837... . - Vaclav Kotesovec, Aug 25 2014
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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:
h:= proc(n) option remember; `if`(n=0, 1, b(n)-(add(b(k)*b(n-k),
k=0..n) -`if`(irem(n, 2)=0, b(n/2), 0))/2)
end:
g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(binomial(h(i), j)*g(n-i*j, i-1), j=0..n/i)))
end:
a:= n-> g(n, n):
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MATHEMATICA
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nn = 20; t[x_] := Sum[a[n] x^n, {n, 1, nn}]; a[0] = 0;
b = Flatten[
sol = SolveAlways[
0 == Series[
t[x] - x Product[1/(1 - x^i)^ a[i], {i, 1, nn}], {x, 0, nn}],
x]; Table[a[n], {n, 0, nn}] /. sol];
r[x_] := Sum[b[[n]] x^(n - 1), {n, 1, nn + 1}]; c =
Drop[CoefficientList[
Series[r[x] - (r[x]^2/2 - r[x^2]/2), {x, 0, nn}], x],
1]; CoefficientList[
Series[Product[(1 + x^i)^c[[i]], {i, 1, nn}], {x, 0, nn}], x] (* Geoffrey Critzer, Nov 15 2014 *)
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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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