From expansion of falling factorials.

Terms

a(0) =4a(1) =15a(2) =52a(3) =151a(4) =372a(5) =799a(6) =1540a(7) =2727a(8) =4516a(9) =7087a(10) =10644a(11) =15415a(12) =21652a(13) =29631a(14) =39652a(15) =52039a(16) =67140a(17) =85327a(18) =106996a(19) =132567a(20) =162484a(21) =197215a(22) =237252a(23) =283111a(24) =335332a(25) =394479a(26) =461140a(27) =535927a(28) =619476a(29) =712447