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%I #58 Sep 27 2022 08:43:44
%S 1,3,24,360,8640,302400,14515200,914457600,73156608000,7242504192000,
%T 869100503040000,124281371934720000,20879270485032960000,
%U 4071457744581427200000,912006534786239692800000,232561666370491121664000000,66977759914701443039232000000
%N a(n) = n!*(n+2)!/2.
%C Also determinant of n X n matrix with m(i,j) = i^2 if i=j, otherwise 1. - _Robert G. Wilson v_, Jan 28 2002
%C Partial products of positive values of A005563. - _Jonathan Vos Post_, Oct 21 2008
%C This sequence has been shown to contain infinitely many squares. From the Hong and Liu abstract: Recently Cilleruelo proved that the product Product_{k=1..n} (k^2 + 1) is a square only for n = 3 which confirms a conjecture of Amdeberhan, Medina and Moll. In this paper, we show that the sequence Product_{k=2..n} (k^2 - 1) contains infinitely many squares. Furthermore, we determine all squares in this sequence. We also give a formula for the p-adic valuation of the terms in this sequence. - _Jonathan Vos Post_, Oct 21 2008
%C Equals (-1)^n * (1, 1, 3, 24, 360, ...) dot (1, -4, 9, -16, 25, ...). E.g., a(4) = (1, 1, 3, 24, 360) dot (1, -4, 9, -16, 25) = 1 - 4 + 27 - 384 + 9000 = 8640. - _Gary W. Adamson_, Apr 21 2009
%H Vincenzo Librandi, <a href="/A010791/b010791.txt">Table of n, a(n) for n = 0..200</a>
%H Javier Cilleruelo, <a href="http://dx.doi.org/10.1016/j.jnt.2007.11.001">Squares in (1^2+1)...(n^2+1)</a>, Journal of Number Theory 128:8 (2008), pp. 2488-2491.
%H Shaofang Hong and Xingjiang Liu, <a href="http://arxiv.org/abs/0810.3366">Squares in (2^2-1)...(n^2-1) and p-adic valuation</a>, arXiv:0810.3366 [math.NT], 2008-2009.
%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.
%F From _Amiram Eldar_, Sep 27 2022: (Start)
%F a(n) = A175430(n+1)/2.
%F Sum_{n>=0} 1/a(n) = 2*BesselI(2,2) = 2*A229020.
%F Sum_{n>=0} (-1)^n/a(n) = 2*BesselJ(2,2). (End)
%p f := n->n!*(n+2)!/2;
%t Table[n!(n+2)!/2,{n,0,20}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 03 2011 *)
%o (Magma) [Factorial(n)* Factorial(n+2) / 2: n in [0..20]]; // _Vincenzo Librandi_, Jun 11 2013
%o (PARI) a(n) = n!*(n+2)!/2; \\ _Michel Marcus_, Feb 03 2016
%Y Cf. A005563, A010792, A010793, A010794, A010795, A175430, A229020.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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