A triangle sequence of the form:q=2:t(n,m,q)=Sum[q^i*Floor[Eulerian[n + 1, m]/2^i], {i, 0, 10}].

Terms

a(0) =1a(1) =1a(2) =1a(3) =1a(4) =12a(5) =1a(6) =1a(7) =37a(8) =37a(9) =1a(10) =1a(11) =116a(12) =452a(13) =116a(14) =1a(15) =1a(16) =305a(17) =2544a(18) =2544a(19) =305a(20) =1a(21) =1a(22) =752a(23) =12497a(24) =25552a(25) =12497a(26) =752a(27) =1a(28) =1a(29) =1761