a(n) = sum(binomial(n,k)*floor(sqrt(Bell(k))),k=0..n).

Terms

a(0) =1a(1) =2a(2) =4a(3) =9a(4) =22a(5) =58a(6) =163a(7) =478a(8) =1439a(9) =4415a(10) =13780a(11) =43757a(12) =141400a(13) =465016a(14) =1555961a(15) =5294885a(16) =18315089a(17) =64357854a(18) =229601019a(19) =831132731a(20) =3051030786a(21) =11351968321a(22) =42788503744a(23) =163309466037a(24) =630861836558