a(n) = (-1)^n*(n+3)!/(2*(n+1)) for n >= 0.

Terms

a(0) =3a(1) =-6a(2) =20a(3) =-90a(4) =504a(5) =-3360a(6) =25920a(7) =-226800a(8) =2217600a(9) =-23950080a(10) =283046400a(11) =-3632428800a(12) =50295168000a(13) =-747242496000