A126224 Determinant of the n X n matrix in which the entries are 1 through n^2, spiraling inward starting with 1 in the (1,1)-entry.

1, -5, -48, 660, 11760, -257040, -6652800, 198918720, 6745939200, -255826771200, -10727081164800, 492775291008000, 24610605962342400, -1327677426915840000, -76940526008586240000, 4766815315895592960000, 314406967644177408000000, -21995911456386651463680000

Offset: 1

Author

Emeric Deutsch, Dec 31 2006

Keywords

• sign

Examples

			For n = 2, the 2 X 2 (spiral) matrix A is
      [1, 2]
      [4, 3]
Then a(2) = -5 because det(A) = 1*3 - 2*4 = -5.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..200
• Sergey Sadov, Problem 11270, American Mathematical Monthly, Vol. 114, No. 1, 2007, p. 78.

Crossrefs

Cf. A023999 (absolute values). - Alois P. Heinz, Jan 21 2014

Programs

Maple

a:=n->(-1)^(n*(n-1)/2)*2^(2*n-3)*(3*n-1)*product(1/2+k,k=0..n-2): seq(a(n),n=1..20);
# second Maple program:
a:= proc(n) option remember; `if`(n<2, (3*n+1)/4,
      4*(1-3*n)*(2*n-5)*(2*n-3) *a(n-2) /(3*n-7))
    end:
seq(a(n), n=1..20);  # Alois P. Heinz, Jan 21 2014

Mathematica

a[n_] := (-1)^(n*(n-1)/2)*2^(2n-3)*(3n-1)*Pochhammer[1/2, n-1]; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, May 26 2015 *)

Formula

a(n) = (-1)^[n*(n-1)/2]*2^(2*n-3)*(3*n-1)*Product_{k=0..n-2} (1/2+k) for n>=2.

E.g.f.: (((-16*x^2-1)*sqrt(2*sqrt(16*x^2+1)+2)-8*sqrt(16*x^2+1)*x^2+16*x^2 + sqrt(16*x^2+1)+1)*sqrt(2*sqrt(16*x^2+1)-2)+(8*(sqrt(16*x^2+1)*x^2+2*x^2-(1/8) * sqrt(16*x^2+1)+1/8))*sqrt(2*sqrt(16*x^2+1)+2))/(512*x^3+32*x). - Robert Israel, Apr 20 2017