A085750 Determinant of the symmetric n X n matrix A defined by A[i,j] = |i-j| for 1 <= i,j <= n.

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

Offset: 1

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 21 2003

Keywords

• easy
• sign

Comments

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Emmanuel Briand, Luis Esquivias, Álvaro Gutiérrez, Adrián Lillo, and Mercedes Rosas, Determinant of the distance matrix of a tree, Proceedings of the 36th Conference on Formal Power Series and Algebraic Combinatorics (Bochum), Séminaire Lotharingien de Combinatoire 91B (2024) Article #29, 12 pp.
• 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).

Crossrefs

Essentially the same as A001787.

Cf. A085807, A100071, A203993, A204249, A278845, A278847.

Programs

Maple

seq((-1)^(n-1)*(n-1)*2^(n-2), n = 1 .. 31);

Mathematica

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 *)

PARI

a(n) = (-1)^n*(1-n)<<(n-2) \\ Charles R Greathouse IV, Sep 30 2022

Formula

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

Extensions

More terms from Philippe Deléham, Nov 16 2008