An antidiagonal triangle based on: t(n,q) = If[n == 0, 1, Sum[Eulerian[n, k]*q^n, {k, 1, n, 2}]].

Terms

a(0) =1a(1) =1a(2) =1a(3) =4a(4) =2a(5) =1a(6) =12a(7) =8a(8) =3a(9) =1a(10) =52a(11) =30a(12) =12a(13) =4a(14) =1a(15) =360a(16) =260a(17) =60a(18) =16a(19) =5a(20) =1a(21) =2656a(22) =2562a(23) =780a(24) =108a(25) =20a(26) =6a(27) =1a(28) =20160a(29) =23408