a(n) is its own 4th difference.

Terms

a(0) =1a(1) =3a(2) =10a(3) =36a(4) =131a(5) =476a(6) =1728a(7) =6272a(8) =22765a(9) =82629a(10) =299915a(11) =1088589a(12) =3951206a(13) =14341527a(14) =52054840a(15) =188941273a(16) =685792227a(17) =2489191330a(18) =9034913540a(19) =32793647355a(20) =119029728628a(21) =432037221840