OFFSET
0,3
COMMENTS
Each row is a cyclic shift to the right by one place of the previous row. See the example below. - N. J. A. Sloane, Jan 07 2019
|a(n)| = number of labeled mappings from n points to themselves (endofunctions) with an odd number of cycles. - Vladeta Jovovic, Mar 30 2006
|a(n)| = number of functions from {1,2,...,n}->{1,2,...,n} such that of all recurrent elements the least is always mapped to the greatest. - Geoffrey Critzer, Aug 29 2013
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..386 (terms n = 1..100 from T. D. Noe)
P. J. Cameron and P. Cara, Independent generating sets and geometries for symmetric groups, J. Algebra, Vol. 258, no. 2 (2002), 641-650.
Sela Fried, Proof of two conjectures on determinants of circulant matrices, 2026.
FORMULA
a(n) = (-1)^(n-1) * n^(n-2) * (n^2 + n)/2 for n>=1, a(0)=1.
E.g.f.[A052182] = E.g.f.[A000312] * E.g.f.[A000272], so A052182(unsigned) is "tree-like". E.g.f.: (T-T^2/2)/(1-T), where T=T(x) is Euler's tree function (see A000169). E.g.f. for signed sequence: (W+W^2/2)/(1+W), where W=W(x)=-T(-x) is the Lambert W function. - Len Smiley, Dec 13 2001
Conjecture: a(n) = -Res( f(n), x^n - 1), where Res is the resultant and f(n) = Sum_{k=1..n} k*x^k. - Benedict W. J. Irwin, Dec 07 2016
Irwin's conjecture is true (see Fried link). - Sela Fried, Jan 26 2026
EXAMPLE
a(3) = 18 because this is the determinant of [(1,2,3), (3,1,2), (2,3,1) ].
MAPLE
seq(LinearAlgebra:-Determinant(Matrix(n, shape=Circulant[[$1..n]])), n=0..30); # Robert Israel, Aug 31 2014
MATHEMATICA
f[n_] := Det[ Table[ RotateLeft[ Range@ n, -j], {j, 0, n - 1}]]; Array[f, 19] (* or *)
f[n_] := (-1)^(n - 1)*n^(n - 2)*(n^2 + n)/2; Array[f, 19]
(* Robert G. Wilson v, Aug 31 2014 *)
Table[Det[Table[RotateRight[Range[k], n], {n, 0, k-1}]], {k, 30}] (* Harvey P. Dale, Jun 20 2024 *)
PROG
(MuPAD) (1+n)^(n-1)*binomial(n+2, n)*(-1)^(n) $ n=0..16 // Zerinvary Lajos, Apr 01 2007
(PARI) a(n) = (n+1)*(-n)^(n-1)/2; \\ Altug Alkan, Dec 17 2017
CROSSREFS
KEYWORD
easy,sign,nice,changed
AUTHOR
Henry M. Gunn High School Mathematical Circle (Joshua Zucker), Jan 26 2000
EXTENSIONS
More terms from James Sellers, Jan 31 2000
a(0)=1 prepended by Alois P. Heinz, Mar 30 2026
STATUS
approved