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Suppose I have a function $f(s,t) = [(1-t^2)(1-s^2t^2)]^{-1/2}$.

Is there a way to get the general coefficient in this power series of the form $s^{2k} t^{2n}$?

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As Artes shows, Series can produce selected coefficients. We can go further: FindSequenceFunction finds closed forms for lists. We have to commit some small sins:

  • We assert that odd coefficients are 0 with opportune [[;; ;;2]]s, not too bad
  • We assert that there are closed forms for each coefficient list $(a_n(k)s^{2k}t^{2n})_n$, and
  • The general coefficient $a(n,k)=a_n(k)$ will admit an obvious closed form.

So we try

FindSequenceFunction@#[[;; ;;2]]&/@CoefficientList[Series[
  1/Sqrt[(1-t^2)(1-s^2t^2)],{s,0,20},{t,0,20}
],{s,t}][[;; ;;2]]

which yields

{Pochhammer[1/2,#1-1]/Pochhammer[1,#1-1]&,
 Pochhammer[1/2,#1-2]/(2Pochhammer[1,#1-2])&,
 3Pochhammer[1/2,#1-3]/(8Pochhammer[1,#1-3])&,...}

where there are a few garbage elements towards the end of the list. This is promising. Type

FindSequenceFunction[{1,1/2,3/8,5/16,35/128,63/256}]

to obtain Pochhammer[1/2,#1-1]/Pochhammer[1,#1-1]&. It would seem

Sum[Pochhammer[1/2,i-n]/Pochhammer[1,i-n]Pochhammer[1/2,n-1]/Pochhammer[1,n-1]
  s^(2(n-1))t^(2(i-1)),{n,1,\[Infinity]},{i,1,\[Infinity]}]

is the series expansion we are looking for. Obviously this method is highly manual. Still, Mathematica was of assistance.

Expanding some definitions and applying some identities, we obtain $$f(s,t)=\sum_{n=0}^\infty\sum_{i=0}^\infty\frac{(-1)^n\prod_{j=1}^i\left(j-n-\frac12\right)}{\Gamma(n+1)\Gamma(1+i-n)}s^{2n}t^{2i}.$$ I think those two $\Gamma$'s can be eliminated, but I'm not very good at complex analysis. I don't know where this is convergent.

Here's a coinciding plot of the two functions

g[s_,t_]=Sum[
  ((-1)^n Product[j-n-1/2,{j,i}])/(Gamma[n+1]Gamma[1+i-n])s^(2n)t^(2i),{n,0,10},{i,0,10}];
Plot3D[{g[s,t],1/Sqrt[(1-t^2)(1-s^2t^2)]},{s,-1,1},{t,-1,1}]
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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

enter image description here

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}
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The standard way to do this would be to simply use SeriesCoefficient, e.g.

SeriesCoefficient[1/Sqrt[1 + t^2], {t, 0, n}]

produces the general form

(I^n (1/2 (-1 + n))!)/(Sqrt[\[Pi]] (n/2)!)

for Mod[n, 2] == 0 && n >= 0. However, it seems to have trouble with your particular function, so you may need to resort to guessing as @Adam has shown...

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