findrecur[f_, ii_, jj_] :=
  (* Finds the recurrence satified by summand f using Sister Celine method. 
        f is a function of two integer arguments, not an expression. 
        ii and jj are the orders of the recurrence that will be found.
        For best results define f externally, using factorials and not binomial
        coefficients. For example, to findrecur for Binomial[n, k]^2, first
        define f[n_, k_]:=n!^2/(k!^2 (n - k)!^2) and then call findrecur[f, 2, 2]  *)
  Block[{a}, 
    Module[{yy, zz, ll, tt, uu, r, s, i, j}, 
      yy = Sum[Sum[
            a[i, j]*FullSimplify[f[n - j, k - i]/f[n, k]], {i, 0, ii}], {j, 0,
             jj}];
      zz = Collect[Numerator[Together[yy]], k];
      ll = CoefficientList[zz, k];
      tt = Flatten[Table[a[i, j], {i, 0, ii}, {j, 0, jj}]];
      uu = Flatten[Simplify[Solve[ll == 0, tt]]];
      For[r = 0, r <= ii, r++, 
        For[s = 0, s <= jj, s++, a[r, s] = Replace[a[r, s], uu]]];
      Sum[Sum[a[i, j] F[n - j, k - i], {i, 0, ii}], {j, 0, jj}] == 0]]