a(n) = binomial(n-h,h)*hypergeometric([h-n/2,h-(n-1)/2],[1],4), h = floor(n/4).

Terms

a(0) =1a(1) =1a(2) =3a(3) =7a(4) =9a(5) =28a(6) =95a(7) =306a(8) =285a(9) =1071a(10) =3948a(11) =14148a(12) =11844a(13) =47160a(14) =182655a(15) =690580a(16) =547965a(17) =2244385a(18) =8961953a(19) =35016345a(20) =26885859a(21) =112052304a(22) =456606332a(23) =1824478488a(24) =1369818996a(25) =5777515212a(26) =23884958520a(27) =97002706248a(28) =71654875560a(29) =304865648208