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

1, -1, -2, -14, -196, -4508, -153272, -7203784, -446634608, -35284134032, -3457845135136, -411483571081184, -58430667093528128, -9757921404619197376, -1893036752496124290944, -422147195806635716880512, -107225387734885472087650048

Offset: 0

Author

Vladimir Reshetnikov, Oct 25 2015

Keywords

• sign
• easy

Examples

			For n = 3, a(3) = (1^2 - 2)*(2^2 - 2)*(3^2 - 2) = -14.
G.f. = 1 - x - 2*x^2 - 14*x^3 - 196*x^4 - 4508*x^5 - 153272*x^6 + ...
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..250

Crossrefs

Cf. A101686, A263688, A263687.

Cf. A008865.

Programs

Haskell

a263766 n = a263766_list !! n
a263766_list = scanl (*) 1 a008865_list
-- Reinhard Zumkeller, Oct 26 2015

Mathematica

Table[Product[k^2 - 2, {k, 1, n}], {n, 0, 16}]
Expand@Table[-Pochhammer[Sqrt[2], n+1] Pochhammer[-Sqrt[2], n+1]/2, {n, 0, 16}]
Join[{1},FoldList[Times,Range[20]^2-2]] (* Harvey P. Dale, Aug 14 2022 *)

PARI

a(n) = prod(k=1, n, k^2-2); \\ Michel Marcus, Oct 25 2015

Formula

a(n) = Gamma(1+sqrt(2)+n)*Gamma(1-sqrt(2)+n)*sin(Pi*sqrt(2))/(Pi*sqrt(2)).

a(n) = A263688(n+1)^2-A263687(n+1)^2/2.

a(n) ~ exp(-2*n)*n^(2*n+1)*sqrt(2)*sin(Pi*sqrt(2)).

G.f. for 1/a(n): hypergeom([1],[1-sqrt(2),1+sqrt(2)], x).

E.g.f. for 1/a(n): hypergeom([],[1-sqrt(2),1+sqrt(2)], x).

E.g.f. for a(n)/n!: hypergeom([1-sqrt(2),1+sqrt(2)], [1], x).

Recurrence: a(0) = 1, a(n) = (n^2-2)*a(n-1).

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