a(n) = S1(n,1), where S1(n, t) = Sum_{k=0..n} (k^t * Sum_{j=0..k} binomial(n,j)).

Terms

a(0) =0a(1) =2a(2) =11a(3) =42a(4) =136a(5) =400a(6) =1104a(7) =2912a(8) =7424a(9) =18432a(10) =44800a(11) =107008a(12) =251904a(13) =585728a(14) =1347584a(15) =3072000a(16) =6946816a(17) =15597568a(18) =34799616a(19) =77201408a(20) =170393600a(21) =374341632a(22) =818937856a(23) =1784676352a(24) =3875536896a(25) =8388608000a(26) =18102616064a(27) =38956695552a(28) =83617644544a(29) =179046449152