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

Terms

a(0) =0a(1) =1a(2) =3a(3) =7a(4) =16a(5) =39a(6) =105a(7) =315a(8) =1048a(9) =3829a(10) =15207a(11) =65071a(12) =297840a(13) =1449755a(14) =7468541a(15) =40555747a(16) =231335960a(17) =1381989881a(18) =8623700811a(19) =56078446615a(20) =379233142800