a(n) = 5^n*sum_{i=1..n} i^5/5^i.

Terms

a(0) =0a(1) =1a(2) =37a(3) =428a(4) =3164a(5) =18945a(6) =102501a(7) =529312a(8) =2679328a(9) =13455689a(10) =67378445a(11) =337053276a(12) =1685515212a(13) =8427947353a(14) =42140274589a(15) =210702132320