a(n) = Sum{0<=k<=n} binomial(n+k,n-k) * k! / (floor(k/2)!)^2.

Terms

a(0) =1a(1) =2a(2) =6a(3) =23a(4) =89a(5) =338a(6) =1286a(7) =4911a(8) =18769a(9) =71722a(10) =273982a(11) =1046119a(12) =3991913a(13) =15222986a(14) =58013678a(15) =220939711a(16) =840883777a(17) =3198349426a(18) =12157775958a(19) =46188298519a(20) =175376312729a(21) =665552754018