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A262396 Product of the sums and differences of the square roots of the first n positive integers, combined in all possible ways.

%I A262396 #19 Jun 18 2022 12:11:45
%S A262396 1,-1,1,64,4096,23323703841,63703464216016403230349121,
%T A262396 316699666163357097153212433469030615484754548657341071360000
%N A262396 Product of the sums and differences of the square roots of the first n positive integers, combined in all possible ways.
%C A262396 The series increases rapidly and the next number in the sequence has 135 decimal digits. Each element for n>1 is necessarily both an integer and a perfect square, the square roots being 1, 8, 64, 152721, 7981444995489, 562760753929551396141111705600, ...
%H A262396 Alois P. Heinz, <a href="/A262396/b262396.txt">Table of n, a(n) for n = 0..10</a>
%e A262396 a(0) = 1 = (empty product).
%e A262396 a(1) = -1 = (sqrt(1)) * (-sqrt(1)).
%e A262396 a(2) = 1 = (1+sqrt(2)) * (1-sqrt(2)) * (-1+sqrt(2)) * (-1-sqrt(2)).
%e A262396 a(3) = 64 = (1+sqrt(2)+sqrt(3)) * (1+sqrt(2)-sqrt(3)) * (1-sqrt(2)+sqrt(3)) * (1-sqrt(2)-sqrt(3)) * (-1+sqrt(2)+sqrt(3)) * (-1+sqrt(2)-sqrt(3)) * (-1-sqrt(2)+sqrt(3)) * (-1-sqrt(2)-sqrt(3)).
%p A262396 s:= proc(n) option remember; `if`(n<2, [1, -1][1..2*n],
%p A262396        map(x-> [x+sqrt(n), x-sqrt(n)][], s(n-1)))
%p A262396     end:
%p A262396 a:= n-> expand(mul(t, t=s(n))):
%p A262396 seq(a(n), n=0..7);  # _Alois P. Heinz_, Sep 21 2015
%t A262396 s[n_] := s[n] = If[n < 2, {1, -1}[[1 ;; 2n]], {# + Sqrt[n], # - Sqrt[n]}& /@ s[n - 1]];
%t A262396 a[n_] := If[n == 0, 1, Times @@ Flatten[s[n], n - 1] // Expand];
%t A262396 a /@ Range[0, 7] (* _Jean-François Alcover_, Nov 24 2020, after _Alois P. Heinz_ *)
%Y A262396 Cf. A354913.
%K A262396 sign
%O A262396 0,4
%A A262396 _Mark Bradley_, Sep 21 2015