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A053605
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Total multiplicity of the eigenvalue 0 in the spectra of the n^(n-2) labeled trees on n vertices.
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0
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1, 0, 3, 8, 135, 1164, 21035, 322832, 7040943, 153153620, 4048737099, 112389077976, 3537768793559, 118535631544316, 4353324736520955, 170245846476629024, 7163230987527864543, 319708454444016133284
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OFFSET
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1,3
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LINKS
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FORMULA
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a(n) = n^(n-1) - 2 * Sum_{m=2..n} (-1)^m * n^(n-m) * m^(m-2)* binomial(n-1, m-1).
G.f. satisfies x^2 + 2*x - x*e^x = Sum_{n >= 1} (a(n)/n!) (x*e^x*e^(-x*e^x))^n.
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MATHEMATICA
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a[n_] := n^(n - 1) - 2*Sum[(-1)^m*n^(n - m)*m^(m - 2)*Binomial[n - 1, m - 1], {m, 2, n}]; Table[a[n], {n, 1, 18}] (* Jean-François Alcover, Dec 10 2012, from formula *)
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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Michel Bauer (bauer(AT)spht.saclay.cea.fr), Jan 20 2000
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EXTENSIONS
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STATUS
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approved
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