a(n) = Sum_{k=0..2*n} (k+1)*T(n, 2*n-k), T given by A027960.

Terms

a(0) =1a(1) =10a(2) =38a(3) =108a(4) =270a(5) =632a(6) =1426a(7) =3148a(8) =6854a(9) =14784a(10) =31674a(11) =67508a(12) =143278a(13) =303016a(14) =638882a(15) =1343388a(16) =2817942a(17) =5898128a(18) =12320650a(19) =25689988a(20) =53477246a(21) =111148920a(22) =230686578a(23) =478150508a(24) =989855590a(25) =2046820192a(26) =4227858266a(27) =8724152148a(28) =17985175374a(29) =37044092744