A179442 - OEIS

%I #14 Oct 23 2023 10:24:50

%S 2,3,16,180,3456,100800,4147200,228614400,16257024000,1448500838400,

%T 158018273280000,20713561989120000,3212195459235840000,

%U 581636820654489600000,121600871304831959040000

%N a(n) = ((n-1)! * (n+1)!) / n.

%F a(n) = Product_{k=1..n} (k * A020725(k)) / (n^2) = Product_{k=1..n} (k * (k+1)) / (n^2).

%F a(n) = A175430(n) / n = A001044(n-1) * (n+1) = ((n -1)^2)! * (n+1).

%F G.f.: 1 + G(0), where G(k)= 1 + x*(k+1)/(1 - (k+2)/(k+2 + 1/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jul 08 2013

%F From _Amiram Eldar_, Jan 18 2021: (Start)

%F Sum_{n>=1} 1/a(n) = BesselI(2,2) + BesselI(3,2) = A229020 + A261879.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = BesselJ(2,2) - BesselJ(3,2). (End)

%e a(5) = ((5-1)! * (5+1)!) / 5 = (4! * 6!) / 5 = (24 * 720) / 5 = 17280 / 5 = 3456.

%e a(5) = ((5 -1)!^2) * (5+1) = 24^2 * 6 = 3456.

%t Table[(n - 1)!*(n + 1)!/n, {n, 1, 15}] (* _Amiram Eldar_, Jan 18 2021 *)

%o (PARI) a(n) = (n-1)!^2*(n+1) \\ _Charles R Greathouse IV_, Oct 23 2023

%Y Cf. A001044, A020725, A175430, A229020, A261879.

%K nonn

%O 1,1

%A _Jaroslav Krizek_, Jul 14 2010