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A085750
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Determinant of the symmetric n X n matrix A defined by A[i,j] = |i-j| for 1 <= i,j <= n.
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18
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0, -1, 4, -12, 32, -80, 192, -448, 1024, -2304, 5120, -11264, 24576, -53248, 114688, -245760, 524288, -1114112, 2359296, -4980736, 10485760, -22020096, 46137344, -96468992, 201326592, -419430400, 872415232, -1811939328, 3758096384, -7784628224, 16106127360
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OFFSET
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1,3
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COMMENTS
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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.]
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
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
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 1..1000
R. L. Graham and H. O. Pollak, On the addressing problem for loop switching, Bell System Tech. J., 50, 1971, 2495-2519.
Tanya Khovanova, Recursive Sequences
R. Merris, The distance spectrum of a tree, J. Graph Theory, 14, No. 3, 1990,365-369.
Index entries for linear recurrences with constant coefficients, signature (-4,-4).
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FORMULA
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a(n) = (-1)^(n+1) * (n-1) * 2^(n-2) = (-1)^(n+1) * A001787(n-1).
G.f.: -x/(1+2x)^2. - Paul Barry, Jan 11 2007
a(n) = -4*a(n-1) - 4*a(n-2); a(1) = 0, a(1) = -1. - Philippe Deléham, Nov 03 2008
E.g.f.: -x*exp(-2*x). - Stefano Spezia, Sep 30 2022
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MAPLE
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seq((-1)^(n-1)*(n-1)*2^(n-2), n = 1 .. 31);
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MATHEMATICA
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Table[-(-1)^n*2^(n - 2)*(n - 1), {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)
LinearRecurrence[{-4, -4}, {0, -1}, 40] (* Harvey P. Dale, Apr 14 2014 *)
CoefficientList[Series[-x/(1 + 2 x)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 15 2014 *)
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PROG
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(PARI) a(n) = (-1)^n*(1-n)<<(n-2) \\ Charles R Greathouse IV, Sep 30 2022
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CROSSREFS
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Essentially the same as A001787.
Cf. A085807, A100071, A203993, A204249, A278845, A278847.
Sequence in context: A260186 A097067 A139756 * A001787 A118442 A038592
Adjacent sequences: A085747 A085748 A085749 * A085751 A085752 A085753
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KEYWORD
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easy,sign
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AUTHOR
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Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 21 2003
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EXTENSIONS
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More terms from Philippe Deléham, Nov 16 2008
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STATUS
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approved
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