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

0, 1, 8, 108, 2304, 72000, 3110400, 177811200, 13005619200, 1185137049600, 131681894400000, 17526860144640000, 2753310393630720000, 504085244567224320000, 106400762391727964160000, 25650183790862991360000000, 7004210187158320840704000000, 2150730290594301893148672000000

Offset

0,3

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.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..200

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

Example

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

Prog

(PARI) a(n) = n!^2*n

Crossrefs

Cf. A002378, A010790, A071951.

Sequence in context: A336828 A184267 A099699 * A138456 A267660 A273057

Adjacent sequences: A084912 A084913 A084914 * A084916 A084917 A084918

Keyword

nonn

Author

Jon Perry, Jul 14 2003

Extensions

a(14) onward from Andrew Howroyd, Oct 29 2025

Status

approved