a(n) is its own 4th difference.

Terms

a(0) =1a(1) =4a(2) =14a(3) =50a(4) =181a(5) =657a(6) =2385a(7) =8657a(8) =31422a(9) =114051a(10) =413966a(11) =1502555a(12) =5453761a(13) =19795288a(14) =71850128a(15) =260791401a(16) =946583628a(17) =3435774958a(18) =12470688498a(19) =45264335853a(20) =164294064481a(21) =596331286321