A213203 The sum of the first n! integers, with every n-th integer taken as negative.

-1, -1, 3, 132, 4260, 172440, 9069480, 609618240, 51209444160, 5267273961600, 651825357321600, 95601055094899200, 16405141092269529600, 3257166195621552614400, 741005309513165913216000

Offset: 1

Author

R. J. Cano, Mar 01 2013

Keywords

• sign
• easy

Examples

			For a(3)=3, 3! is 6 then the sum of the first 6 integers taking each 3rd integer as negative is: 1+2-3+4+5-6 = 3.
For a(4)=132, 4! is 24 then the sum of the first 24 integers taking each 4th integer as negative is: 1+2+3-4+5+6+7-8+9+10+11-12+13+14+15-16+17+18+19-20+21+22+23-24 = 132.
		

Links

• R. J. Cano, Table of n, a(n) for n = 1..60
• R. J. Cano, Additional information

Crossrefs

Cf. A181482, A055555.

Programs

PARI

a(n)={my(y=(n-1)!);((n*y)*((n-2)*y-1))\2;}

Formula

a(n) = n * (n-1)! * ((n-2)*(n-1)! - 1)/2.

Conjecture: a(n) + (-n^2-n-11)*a(n-1) + (n^3+7*n^2-13*n+39)*a(n-2) - 2*(n-2)*(4*n^2-2*n-15)*a(n-3) + 20*(n-2)*(n-3)*(n-4)*a(n-4) = 0. - R. J. Mathar, Mar 21 2013