%I #6 Jun 06 2021 19:22:12

%S 2,2,6,16,65,312,1813,12288,95617,840960,8254081,89441280,1060369921,

%T 13649610240,189550368001,2824077312000,44927447040001,

%U 760034451456000,13622700994560001,257872110354432000,5140559166898176001

%N a(n) = Sum_{k=1..n} ((n-k)!*(k-1)! - (-1)^k).

%C According to the Generalized Wilson-Lagrange Theorem, a prime p divides (p-k)!*(k-1)! - (-1)^k for all integers k > 0. p divides a(p) for prime p. Quotients a(p)/p are listed in A134296(n) = {1, 2, 13, 259, 750371, 81566917, 2642791002353, 716984262871579, 102688143363690674087, ...}. p^2 divides a(p) for prime p = {7, 71}.

%F a(n) = Sum_{k=1..n} ((n-k)!*(k-1)! - (-1)^k).

%t Table[ Sum[ (n-k)!*(k-1)! - (-1)^k, {k,1,n} ], {n,1,30} ]

%Y Cf. A007540, A007619 (Wilson quotients: ((p-1)!+1)/p).

%Y Cf. A134296 (quotients a(p)/p).

%K nonn

%O 1,1

%A _Alexander Adamchuk_, Oct 17 2007