a(n) = Sum_{k=1..n} binomial(n,k) * tau(k) * tau(n - k + 1), where tau = A000005.

Terms

a(0) =1a(1) =6a(2) =20a(3) =55a(4) =142a(5) =322a(6) =779a(7) =1608a(8) =3894a(9) =7370a(10) =18372a(11) =33137a(12) =81512a(13) =149694a(14) =353224a(15) =641461a(16) =1570836a(17) =2684928a(18) =6642915a(19) =11795178a(20) =28133846a(21) =46768200a(22) =125433400a(23) =197654545a(24) =485749918a(25) =893864394a(26) =2066417482a(27) =3385115393a(28) =8975476976a(29) =14384181908