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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)] &

We have defined cf to get only even coefficients (2k), (2n), since the others do vanish.

e.g.

cf[4, 6]

105/1024

Table[cf[k, n], {k, 6}, {n, 6}] // MatrixForm

One can get the same with Array as well

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

True

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)] &

We have defined cf to get only even coefficients (2k,2n), since the others do vanish.

Table[cf[k, n], {k, 6}, {n, 6}] // MatrixForm

One can get the same with Array as well

Array[ cf, {6, 6}] == Table[ cf[k, n], {k, 6}, {n, 6}]

True

This appraoch would be satisfactory for small {n, k}. For bigger ones there is SeriesCoefficient (as another answer recalled).

sf[k_, n_]:= SeriesCoefficient[ 1/Sqrt[(1 - t^2) (1 - s^2 t^2)], {s, 0, 2k}, {t, 0, 2n}]

Array[ cf, {6, 6}] == Array[ sf, {6, 6}]

True

And sf is much better than cf:

AbsoluteTiming[sf[350, 400];]

{0.0015182, Null}

AbsoluteTiming[cf[350, 400];]

{10.2873, Null}

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

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)] &

We have defined cf to get only even coefficients (2k), (2n), since the others do vanish.

e.g.

cf[4, 6]

105/1024

Table[cf[k, n], {k, 6}, {n, 6}] // MatrixForm

One can get the same with Array as well

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

True

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

cf[k_, n_] := Series[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]

5/1024

Table[ cf[k, n], {k, 6}, {n, 6}]

{{-(1/2), 1/4, 1/16, 1/32, 5/256, 7/512}, 
 {0, -(1/8), 1/16, 1/64, 1/128, 5/1024}, 
 {0, 0, -(1/16), 1/32, 1/128, 1/256}, 
 {0, 0, 0, -(5/128), 5/256, 5/1024}, 
 {0, 0, 0, 0, -(7/256), 7/512}, 
 {0, 0, 0, 0, 0, -(21/1024)}} 

One can get the same with Array as well

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

True

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