login
The sum of the first n! integers, with every n-th integer taken as negative.
4

%I #23 Mar 21 2013 16:50:08

%S -1,-1,3,132,4260,172440,9069480,609618240,51209444160,5267273961600,

%T 651825357321600,95601055094899200,16405141092269529600,

%U 3257166195621552614400,741005309513165913216000

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

%H R. J. Cano, <a href="/A213203/b213203.txt">Table of n, a(n) for n = 1..60</a>

%H R. J. Cano, <a href="/wiki/First_sum_everything_then_subtract_twice">Additional information</a>

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

%F 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

%e 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.

%e 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.

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

%Y Cf. A181482, A055555.

%K sign,easy

%O 1,3

%A _R. J. Cano_, Mar 01 2013