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A263766 a(n) = Product_{k=1..n} (k^2 - 2).

%I A263766 #23 Nov 16 2024 18:49:36
%S A263766 1,-1,-2,-14,-196,-4508,-153272,-7203784,-446634608,-35284134032,
%T A263766 -3457845135136,-411483571081184,-58430667093528128,
%U A263766 -9757921404619197376,-1893036752496124290944,-422147195806635716880512,-107225387734885472087650048
%N A263766 a(n) = Product_{k=1..n} (k^2 - 2).
%H A263766 Reinhard Zumkeller, <a href="/A263766/b263766.txt">Table of n, a(n) for n = 0..250</a>
%F A263766 a(n) = Gamma(1+sqrt(2)+n)*Gamma(1-sqrt(2)+n)*sin(Pi*sqrt(2))/(Pi*sqrt(2)).
%F A263766 a(n) = A263688(n+1)^2-A263687(n+1)^2/2.
%F A263766 a(n) ~ exp(-2*n)*n^(2*n+1)*sqrt(2)*sin(Pi*sqrt(2)).
%F A263766 G.f. for 1/a(n): hypergeom([1],[1-sqrt(2),1+sqrt(2)], x).
%F A263766 E.g.f. for 1/a(n): hypergeom([],[1-sqrt(2),1+sqrt(2)], x).
%F A263766 E.g.f. for a(n)/n!: hypergeom([1-sqrt(2),1+sqrt(2)], [1], x).
%F A263766 Recurrence: a(0) = 1, a(n) = (n^2-2)*a(n-1).
%F A263766 0 = a(n)*(-24*a(n+2) - 15*a(n+3) + a(n+4)) + a(n+1)*(-9*a(n+2) - 4*a(n+3)) + a(n+2)*(+3*a(n+2)) if n>=0. - _Michael Somos_, Oct 30 2015
%e A263766 For n = 3, a(3) = (1^2 - 2)*(2^2 - 2)*(3^2 - 2) = -14.
%e A263766 G.f. = 1 - x - 2*x^2 - 14*x^3 - 196*x^4 - 4508*x^5 - 153272*x^6 + ...
%t A263766 Table[Product[k^2 - 2, {k, 1, n}], {n, 0, 16}]
%t A263766 Expand@Table[-Pochhammer[Sqrt[2], n+1] Pochhammer[-Sqrt[2], n+1]/2, {n, 0, 16}]
%t A263766 Join[{1},FoldList[Times,Range[20]^2-2]] (* _Harvey P. Dale_, Aug 14 2022 *)
%o A263766 (PARI) a(n) = prod(k=1, n, k^2-2); \\ _Michel Marcus_, Oct 25 2015
%o A263766 (Haskell)
%o A263766 a263766 n = a263766_list !! n
%o A263766 a263766_list = scanl (*) 1 a008865_list
%o A263766 -- _Reinhard Zumkeller_, Oct 26 2015
%Y A263766 Cf. A101686, A263688, A263687.
%Y A263766 Cf. A008865.
%K A263766 sign,easy
%O A263766 0,3
%A A263766 _Vladimir Reshetnikov_, Oct 25 2015