a(n) = 1 + F(2*n+1) - (F(n+4) - (-1)^n*F(n-2))/2 where F=A000045.

Terms

a(0) =0a(1) =0a(2) =2a(3) =7a(4) =25a(5) =72a(6) =208a(7) =564a(8) =1530a(9) =4059a(10) =10769a(11) =28336a(12) =74560a(13) =195576a(14) =513010a(15) =1344063a(16) =3521385a(17) =9221688a(18) =24149456a(19) =63230860a(20) =165558250a(21) =433454835a(22) =1134845857a(23) =2971111392a(24) =7778592000a(25) =20364739632a(26) =53315898338a(27) =139583151799a(28) =365434267705a(29) =956720165544