a(n) = (Sum_{k=1..n} k^3 * p(k) * p(n-k)) * 2/n where p = A000041.

Terms

a(0) =2a(1) =17a(2) =66a(3) =218a(4) =564a(5) =1407a(6) =3074a(7) =6536a(8) =12960a(9) =24991a(10) =46028a(11) =83166a(12) =145182a(13) =249151a(14) =417432a(15) =688742a(16) =1114978a(17) =1782228a(18) =2804886a(19) =4365560a(20) =6709074a(21) =10208894a(22) =15368870a(23) =22937184a(24) =33916388a(25) =49763175a(26) =72429396a(27) =104685554a(28) =150234266a(29) =214249998