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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.

%I A126224 #26 Apr 07 2021 20:05:33
%S A126224 1,-5,-48,660,11760,-257040,-6652800,198918720,6745939200,
%T A126224 -255826771200,-10727081164800,492775291008000,24610605962342400,
%U A126224 -1327677426915840000,-76940526008586240000,4766815315895592960000,314406967644177408000000,-21995911456386651463680000
%N 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.
%H A126224 Alois P. Heinz, <a href="/A126224/b126224.txt">Table of n, a(n) for n = 1..200</a>
%H A126224 Sergey Sadov, <a href="http://www.jstor.org/stable/27642128">Problem 11270</a>, American Mathematical Monthly, Vol. 114, No. 1, 2007, p. 78.
%F A126224 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.
%F A126224 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
%e A126224 For n = 2, the 2 X 2 (spiral) matrix A is
%e A126224       [1, 2]
%e A126224       [4, 3]
%e A126224 Then a(2) = -5 because det(A) = 1*3 - 2*4 = -5.
%p A126224 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);
%p A126224 # second Maple program:
%p A126224 a:= proc(n) option remember; `if`(n<2, (3*n+1)/4,
%p A126224       4*(1-3*n)*(2*n-5)*(2*n-3) *a(n-2) /(3*n-7))
%p A126224     end:
%p A126224 seq(a(n), n=1..20);  # _Alois P. Heinz_, Jan 21 2014
%t A126224 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 *)
%Y A126224 Cf. A023999 (absolute values). - _Alois P. Heinz_, Jan 21 2014
%K A126224 sign
%O A126224 1,2
%A A126224 _Emeric Deutsch_, Dec 31 2006