a(n) = Sum_{k=0..n} (k+1)*T(n, n-k), where T is given by A008288.

Terms

a(0) =1a(1) =3a(2) =10a(3) =30a(4) =87a(5) =245a(6) =676a(7) =1836a(8) =4925a(9) =13079a(10) =34446a(11) =90090a(12) =234227a(13) =605865a(14) =1560200a(15) =4002072a(16) =10230201a(17) =26069995a(18) =66251090a(19) =167941494a(20) =424753615a(21) =1072057117a(22) =2700704172a(23) =6791746500a(24) =17052595573a(25) =42752015487a(26) =107035180630a(27) =267634562754a(28) =668407232235