clearvars, clc

digits(5000)

f=vpa(1); nxtno=2;

for n=1:11

   rhs=p2(n); 

   disp([n rhs]) 

   while f<rhs

      f=f*nxtno; nxtno=nxtno+1;

   end

   if rhs==f

      disp(f) 

   end

end

function pr = p2(n)

 

  p0=vpa(2)^vpa(n);

  p=vpa(1);

  for i=p0-1:-1:p0-p0/2

     p=p*vpa(i); 

  end

  pr=p;

end

finds only n=1 and n=2, where the respective equal values are 1 and 6. The limit of testing was n=11 as beyond that the RHS becomes larger than the allotted 5000 digits (the program keeps building the factorial until it equals or exceeds the RHS).

Oh, yes, forgot: the k values are 1 and 3 respectively.