a(n) = n*F(n+1) - (n+1)*F(n), where F = A000045.

Terms

a(0) =0a(1) =-1a(2) =1a(3) =1a(4) =5a(5) =10a(6) =22a(7) =43a(8) =83a(9) =155a(10) =285a(11) =516a(12) =924a(13) =1639a(14) =2885a(15) =5045a(16) =8773a(17) =15182a(18) =26162a(19) =44915a(20) =76855a(21) =131119a(22) =223101a(23) =378696a(24) =641400a(25) =1084175a(26) =1829257a(27) =3081193a(28) =5181893a(29) =8702290