Triangle T(n, k) = n!*Sum_{j=k..n} (-1)^(j+k)*binomial(k+j, j)/j!, read by rows.

Terms

a(0) =1a(1) =0a(2) =2a(3) =1a(4) =1a(5) =6a(6) =2a(7) =7a(8) =8a(9) =20a(10) =9a(11) =23a(12) =47a(13) =45a(14) =70a(15) =44a(16) =121a(17) =214a(18) =281a(19) =224a(20) =252a(21) =265a(22) =719a(23) =1312a(24) =1602a(25) =1554a(26) =1050a(27) =924a(28) =1854a(29) =5041