a(n) = Sum_{k=0..n} binomial(n+k+2,n-k) * Fibonacci(k+1).

Terms

a(0) =1a(1) =4a(2) =13a(3) =42a(4) =138a(5) =457a(6) =1516a(7) =5029a(8) =16680a(9) =55320a(10) =183469a(11) =608476a(12) =2018017a(13) =6692778a(14) =22196682a(15) =73615573a(16) =244146964a(17) =809716441a(18) =2685434640a(19) =8906277360a(20) =29537779561a(21) =97962413044a(22) =324893560453