In Singular:

  > ring 0,(x,y),dp;
  > LIB "rinvar.lib";
  > poly f = y6-y2-x2;
  > hilb(f, 1);
  1,0,0,0,0,0,-1,0

This means the Hilbert polynomial is 1-t^6. 
Then in Maple

  > series((1-t^6)/(1-t)^3,t=0);
  1 + 3t + 6t^2 + 10t^3 + 15t^4 + 21t^5 + O(t^6)

This is the Hilbert series.

That's it.