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A263766 a(n) = Product_{k=1..n} (k^2 - 2). 2
1, -1, -2, -14, -196, -4508, -153272, -7203784, -446634608, -35284134032, -3457845135136, -411483571081184, -58430667093528128, -9757921404619197376, -1893036752496124290944, -422147195806635716880512, -107225387734885472087650048 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

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

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

EXAMPLE

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 + ...

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}]

PROG

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

(Haskell)

a263766 n = a263766_list !! n

a263766_list = scanl (*) 1 a008865_list

-- Reinhard Zumkeller, Oct 26 2015

CROSSREFS

Cf. A101686, A263688, A263687.

Cf. A008865.

Sequence in context: A047796 A305112 A232686 * A244577 A090300 A213977

Adjacent sequences:  A263763 A263764 A263765 * A263767 A263768 A263769

KEYWORD

sign,easy

AUTHOR

Vladimir Reshetnikov, Oct 25 2015

STATUS

approved

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Last modified September 17 10:52 EDT 2019. Contains 327129 sequences. (Running on oeis4.)