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A204044 Lozenge Gaussian integer factorial, product of all nonzero Gaussian integers a + bi for which |a| + |b| <= n. 2

%I

%S -1,64,-3240000,530841600000000,-791432829997056000000000000,

%T 24298387172648346846064803840000000000000000,

%U -30208456145049398593072092383690495361024000000000000000000000000

%N Lozenge Gaussian integer factorial, product of all nonzero Gaussian integers a + bi for which |a| + |b| <= n.

%C Called "lozenge" because the Gaussian integers considered here form up a lozenge or diamond in the complex plane.

%C All terms are purely real integers.

%F log |a(n)| ~ 2n^2 log n. - _Charles R Greathouse IV_, May 01 2012

%e a(2) = -2 * (-1 - i) * -1 * (-1 + i) * -2i * -i * i * 2i * (1 - i) * 1 * (1 + i) * 2 = 64. (Note that numbers like -2 + i are skipped over in the computation of a(2) because abs(-2) + abs(1) > 2).

%t Table[Times@@Select[ReplaceAll[Flatten[Table[a + b I, {a, -n, n}, {b, -n, n}]], 0 -> 1], Abs[Re[#]] + Abs[Im[#]] <= n&], {n, 10}]

%o (PARI) a(n)=(-1)^n*prod(i=1,n,prod(j=1,n-i,i^2+j^2))^2*n!^4 \\ _Charles R Greathouse IV_, May 01 2012

%Y Cf. A204041, square factorial; A204043, circle factorial.

%K sign

%O 1,2

%A _Alonso del Arte_, Jan 09 2012

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Last modified May 17 09:20 EDT 2022. Contains 353741 sequences. (Running on oeis4.)