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Total multiplicity of the eigenvalue 0 in the spectra of the n^(n-2) labeled trees on n vertices.
0

%I #24 Jul 22 2019 16:19:01

%S 1,0,3,8,135,1164,21035,322832,7040943,153153620,4048737099,

%T 112389077976,3537768793559,118535631544316,4353324736520955,

%U 170245846476629024,7163230987527864543,319708454444016133284

%N Total multiplicity of the eigenvalue 0 in the spectra of the n^(n-2) labeled trees on n vertices.

%H M. Bauer and O. Golinelli, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/BAUER/zerotree.html">On the kernel of tree incidence matrices</a>, J. Integer Sequences, Vol. 3 (2000), #00.1.4.

%F a(n) = n^(n-1) - 2 * Sum_{m=2..n} (-1)^m * n^(n-m) * m^(m-2)* binomial(n-1, m-1).

%F G.f. satisfies x^2 + 2*x - x*e^x = Sum_{n >= 1} (a(n)/n!) (x*e^x*e^(-x*e^x))^n.

%t 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 *)

%K nonn,easy,nice

%O 1,3

%A Michel Bauer (bauer(AT)spht.saclay.cea.fr), Jan 20 2000

%E More terms from _David W. Wilson_, Dec 08 2000