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

Terms

a(0) =1a(1) =2a(2) =6a(3) =19a(4) =65a(5) =231a(6) =841a(7) =3110a(8) =11628a(9) =43834a(10) =166298a(11) =634140a(12) =2428336a(13) =9331688a(14) =35967462a(15) =138987715a(16) =538287881a(17) =2088842463a(18) =8119916647a(19) =31613327405a(20) =123251518641a(21) =481125828853