a(n) = Sum_{k=1..n} binomial(n,k) * sigma(k) * sigma(n - k + 1), where sigma = A000203.

Terms

a(0) =1a(1) =9a(2) =43a(3) =155a(4) =511a(5) =1442a(6) =4131a(7) =10323a(8) =28171a(9) =63987a(10) =171667a(11) =369395a(12) =957958a(13) =2047694a(14) =5078963a(15) =10671529a(16) =26542339a(17) =53522031a(18) =132273403a(19) =268623854a(20) =647842889a(21) =1266118858a(22) =3197923083a(23) =6058756355a(24) =14581380971a(25) =29480406552a(26) =68634048862a(27) =131847974143a(28) =323289015466a(29) =611887749996