Array read by antidiagonals: T(m,n) = number of nonzero m X n binary arrays with all 1's connected.

Terms

a(0) =1a(1) =3a(2) =3a(3) =6a(4) =13a(5) =6a(6) =10a(7) =40a(8) =40a(9) =10a(10) =15a(11) =108a(12) =218a(13) =108a(14) =15a(15) =21a(16) =275a(17) =1126a(18) =1126a(19) =275a(20) =21a(21) =28a(22) =681a(23) =5726a(24) =11506a(25) =5726a(26) =681a(27) =28a(28) =36a(29) =1664