a(n) = Sum_{k=0..floor(n/4)} (-1)^k * binomial(n+3,4*k+3) * Catalan(k).

Terms

a(0) =1a(1) =4a(2) =10a(3) =20a(4) =34a(5) =48a(6) =48a(7) =0a(8) =-163a(9) =-548a(10) =-1274a(11) =-2340a(12) =-3255a(13) =-2224a(14) =5304a(15) =28560a(16) =82379a(17) =182300a(18) =322102a(19) =410700a(20) =133128a(21) =-1295264a(22) =-5440600a(23) =-14733680a(24) =-31384533a(25) =-52870668a(26) =-59633454a(27) =11449780a(28) =312532426a(29) =1137823168