A084915 - OEIS

%I #17 May 07 2019 08:36:23

%S 0,1,8,108,2304,72000,3110400,177811200,13005619200,1185137049600,

%T 131681894400000,17526860144640000,2753310393630720000,

%U 504085244567224320000

%N a(n) = (n!)^2*n.

%C Used to prove that Sum_{n>=1} 1/A002378(n) = 1. Examining Sum_{n=1..k} 1/A002378(n) gives 1/2, 1/2 + 1/6, 1/2 + 1/6 + 1/12. Simplifying gives 1/2, 8/12, 108/144, where the numerators are this sequence and the denominators are A010790. Therefore we have k!^2*k/k!(k+1)! = k*k!/(k+1)! = k/(k+1), which tends to 1 as k tends to infinity.

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

%F a(n) = det(PS(i+2,j+1), 1 <= i,j <= n-1), where PS(n,k) are Legendre-Stirling numbers of the second kind (A071951) and n > 0. [_Mircea Merca_, Apr 06 2013]

%e a(3) = 3!^2*3 = 36*3 = 108.

%o (PARI) for(n=1,50,print1(n!^2*n","))

%Y Cf. A002378, A010790, A071951.

%K nonn

%O 0,3

%A _Jon Perry_, Jul 14 2003