Square array read by antidiagonals, A(n,k) = k!*[x^k]((1-Sum_{j=1..n} x^j)^(-1)), (n>=0,k>=0).

Terms

a(0) =1a(1) =1a(2) =0a(3) =1a(4) =1a(5) =0a(6) =1a(7) =1a(8) =2a(9) =0a(10) =1a(11) =1a(12) =4a(13) =6a(14) =0a(15) =1a(16) =1a(17) =4a(18) =18a(19) =24a(20) =0a(21) =1a(22) =1a(23) =4a(24) =24a(25) =120a(26) =120a(27) =0a(28) =1a(29) =1