a(n) = S1(n,2), where S1(n, t) = Sum_{k=0..n} (k^t * Sum_{j=0..k} binomial(n,j)).

Terms

a(0) =0a(1) =2a(2) =19a(3) =104a(4) =440a(5) =1600a(6) =5264a(7) =16128a(8) =46848a(9) =130560a(10) =352000a(11) =923648a(12) =2369536a(13) =5963776a(14) =14766080a(15) =36044800a(16) =86900736a(17) =207224832a(18) =489357312a(19) =1145569280a(20) =2660761600a(21) =6136266752a(22) =14060355584a(23) =32027705344a(24) =72561459200a(25) =163577856000a(26) =367068708864a(27) =820204535808