A084915 a(n) = (n!)^2*n.

0, 1, 8, 108, 2304, 72000, 3110400, 177811200, 13005619200, 1185137049600, 131681894400000, 17526860144640000, 2753310393630720000, 504085244567224320000

Offset: 0

Author

Jon Perry, Jul 14 2003

Keywords

• nonn

Comments

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.

Examples

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

Crossrefs

Cf. A002378, A010790, A071951.

Programs

PARI

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

Formula

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

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]