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%I #50 Sep 30 2022 16:23:06
%S 0,-1,4,-12,32,-80,192,-448,1024,-2304,5120,-11264,24576,-53248,
%T 114688,-245760,524288,-1114112,2359296,-4980736,10485760,-22020096,
%U 46137344,-96468992,201326592,-419430400,872415232,-1811939328,3758096384,-7784628224,16106127360
%N Determinant of the symmetric n X n matrix A defined by A[i,j] = |i-j| for 1 <= i,j <= n.
%C The determinant of the distance matrix of a tree with vertex set {1,2,...,n}. The distance matrix is the n X n matrix in which the (i,j)-term is the number of edges in the unique path from vertex i to vertex j. [The matrix A in the definition is the distance matrix of the path-tree 1-2-...-n.]
%C Hankel transform of A100071. Also Hankel transform of C(2n-2,n-1)(-1)^(n-1). Inverse binomial transform of -n. - _Paul Barry_, Jan 11 2007
%C Pisano period lengths: 1, 1, 3, 1, 20, 3, 42, 1, 9, 20, 55, 3,156, 42, 60, 1,136, 9,171, 20, ... - _R. J. Mathar_, Aug 10 2012
%H Vincenzo Librandi, <a href="/A085750/b085750.txt">Table of n, a(n) for n = 1..1000</a>
%H R. L. Graham and H. O. Pollak, <a href="http://www.math.ucsd.edu/~ronspubs/71_05_loop_switching.pdf">On the addressing problem for loop switching</a>, Bell System Tech. J., 50, 1971, 2495-2519.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H R. Merris, <a href="http://dx.doi.org/10.1002/jgt.3190140309">The distance spectrum of a tree</a>, J. Graph Theory, 14, No. 3, 1990,365-369.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-4,-4).
%F a(n) = (-1)^(n+1) * (n-1) * 2^(n-2) = (-1)^(n+1) * A001787(n-1).
%F G.f.: -x/(1+2x)^2. - _Paul Barry_, Jan 11 2007
%F a(n) = -4*a(n-1) - 4*a(n-2); a(1) = 0, a(1) = -1. - _Philippe Deléham_, Nov 03 2008
%F E.g.f.: -x*exp(-2*x). - _Stefano Spezia_, Sep 30 2022
%p seq((-1)^(n-1)*(n-1)*2^(n-2), n = 1 .. 31);
%t Table[-(-1)^n*2^(n - 2)*(n - 1), {n, 1, 50}] (* _Vladimir Joseph Stephan Orlovsky_, Jun 10 2011 *)
%t LinearRecurrence[{-4,-4},{0,-1},40] (* _Harvey P. Dale_, Apr 14 2014 *)
%t CoefficientList[Series[-x/(1 + 2 x)^2, {x, 0, 40}], x] (* _Vincenzo Librandi_, Apr 15 2014 *)
%o (PARI) a(n) = (-1)^n*(1-n)<<(n-2) \\ _Charles R Greathouse IV_, Sep 30 2022
%Y Essentially the same as A001787.
%Y Cf. A085807, A100071, A203993, A204249, A278845, A278847.
%K easy,sign
%O 1,3
%A Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 21 2003
%E More terms from _Philippe Deléham_, Nov 16 2008
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