a(n) = -Sum_{0<=i<j<=n} (-1)^(i+j) * (i+j)!/(i!*j!).

Terms

a(0) =0a(1) =1a(2) =3a(3) =10a(4) =34a(5) =120a(6) =434a(7) =1597a(8) =5949a(9) =22363a(10) =84655a(11) =322245a(12) =1232205a(13) =4729453a(14) =18210279a(15) =70307546a(16) =272087770a(17) =1055139408a(18) =4099200524a(19) =15951053566a(20) =62159391150a(21) =242542955378a(22) =947504851414