Implicitly assuming that one deals with series expansion around $(s,t)=(0,0)$ we can define

    cf[k_, n_] := 
      Series[1/Sqrt[(1 - t^2) (1 - s^2 t^2)], {s, 0, 2 k}, {t, 0, 2 n}] // 
      Normal // Coefficient[#, s^(2 k) t^(2 n)] &

e.g.

    cf[4, 6]
>     105/1024

    Table[ cf[k, n], {k, 6}, {n, 6}]
>     {{1/2, 1/4, 3/16, 5/32, 35/256, 63/512}, 
>      {0, 3/8, 3/16, 9/64, 15/128, 105/1024}, 
>      {0, 0, 5/16, 5/32, 15/128, 25/256}, 
>      {0, 0, 0, 35/128, 35/256, 105/1024}, 
>      {0, 0, 0, 0, 63/256, 63/512},
>      {0, 0, 0, 0, 0, 231/1024}}
One can get the same with Array as well

    Array[cf[#1, #2] &, {6, 6}] == Table[cf[k, n], {k, 6}, {n, 6}]
>     True