%I #6 Jan 27 2025 21:13:10
%S 0,1,3,18,94,611,4381,35834,326662,3301193,36611767,442379438,
%T 5784595282,81393560783,1226280162097,19696507698878,335990910075202,
%U 6066382761193373,115578717461844067,2317323290060066858
%N a(n)= n! - (n-1)!! - a(n-1), with a(1)=1!-0!!=0.
%F a(n)= n! - (n-1)!! - a(n-1), with a(1)=1!-0!!=0. a(n)=abs{sum{i=1..n,[(n-i+1)!-(n-i)!! ]*(-1)^i}}
%e a(1)=1!-0!!=1-1=0
%e a(2)=2!-1!!-1!+0!!=2-1-1+1=1
%e a(3)=3!-2!!-2!+1!!+1!-0!!=6-2-2+1+1-1=3
%e a(4)=4!-3!!-3!+2!!+2!-1!!-1!+0!!=24-3-6+2+2-1-1+1=18
%e a(5)=5!-4!!-4!+3!!+3!-2!!-2!+1!!+1!-0!!=120-8-24+3+6-2-2+1+1-1=94
%p P:=proc(n) local a,i,k,w; print(0); a:=0; for i from 2 by 1 to n do k:=i-1; w:=i-3; while w>0 do k:=k*w; w:=w-2; od; a:=(i!-k)-a; print(a); od; end: P(50);
%t a[1]=0;a[n_]:=n!-(n-1)!!-a[n-1];Table[a[n],{n,20}] (* _James C. McMahon_, Jan 27 2025 *)
%K nonn
%O 0,3
%A _Paolo P. Lava_ and _Giorgio Balzarotti_, May 30 2007