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A217420
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Number of rooted unlabeled trees where the root node has degree 2 and both branches are distinct.
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2
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0, 0, 0, 1, 2, 6, 14, 37, 92, 239, 613, 1607, 4215, 11185, 29814, 80070, 216061, 586218, 1597292, 4370721, 12003163, 33077327, 91431425, 253454781, 704425087, 1962537755, 5479843060, 15332668869, 42983623237, 120716987723, 339595975795, 956840683968
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OFFSET
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1,5
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REFERENCES
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F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973, page 57.
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LINKS
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FORMULA
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O.g.f.: x * (T(x)^2/2 - T(x^2)/2) where T(x) is o.g.f. for A000081.
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MAPLE
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with(numtheory):
b:= proc(n) option remember; `if`(n<=1, n,
(add(add(d*b(d), d=divisors(j))*b(n-j), j=1..n-1))/(n-1))
end:
a:= proc(n) option remember; (add(b(k)*b(n-1-k), k=0..n-1)-
`if`(irem(n, 2, 'r')=1, b(r), 0))/2
end:
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MATHEMATICA
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Needs["Combinatorica`"]
nn=30; s[n_, k_]:=s[n, k]=a[n+1-k]+If[n<2k, 0, s[n-k, k]]; a[1]=1; a[n_]:=a[n]=Sum[a[i]s[n-1, i]i, {i, 1, n-1}]/(n-1); rt=Table[a[i], {i, 1, nn}]; Take[CoefficientList[CycleIndex[AlternatingGroup[2], s]-CycleIndex[SymmetricGroup[2], s]/.Table[s[j]->Table[Sum[rt[[i]]x^(i*k), {i, 1, nn}], {k, 1, nn}][[j]], {j, 1, nn}], x], nn] (* after code by Robert A. Russell in A000081 *)
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PROG
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(Python)
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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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