RR := RealField(6);
r:=6;

X:= t^3*((1-t^(r-2)) div (1-t)) * ((1-t^r) div (1-t));
Y:= -t^(r+1);
A:=(Y+1/r*X)/&*[1-t^i: i in [1,1,r,r]];
[RR!(r^2*Coefficient(S!A,i)/(i^3)) : i in [380..399]];

s5:= 3;
s6:=2;

[RR!(2*r^2*Coefficient(S!A,i)/(i^3)) : i in [380..399]];

should give [1,1,1,1..] 

A:=(Y+1/r*X)/&*[1-t^i: i in [1,1,r,r]] - 1/r^2*(t+4*t^2+t^3)/(1-t)^4;

gives almost zero e-5 stuff (exponential notation)

s*r^2*(1-t^r)^2*A  // is now sensible expression for r = 11

-4*t^21-32*t^20-75*t^19-124*t^18-170*t^17-204*t^16
 -217*t^15-200*t^14-144*t^13-40*t^12+121*t^11
  -40*t^10-144*t^9-200*t^8-217*t^7-204*t^6-170*t^5-124*t^4-75*t^3-32*t^2-4*t

for r = 5 get s = 3 and numerator =
-t^9 - 8*t^8 - 12*t^7 - 4*t^6 + 25*t^5 - 4*t^4 - 12*t^3 - 8*t^2 - t

for r = 6 get s = 2 and numerator =
 -t^11-8*t^10-15*t^9-16*t^8-5*t^7+24*t^6-5*t^5-16*t^4-15*t^3-8*t^2-t

for r = 7 get s = 3/2 and num =
-t^13-8*t^12-33/2*t^11-22*t^10-20*t^9-6*t^8+49/2*t^7
 -6*t^6-20*t^5-22*t^4-33/2*t^3-8*t^2-t

for r = 8 get s = 6/5 and num =
-t^15-8*t^14-87/5*t^13-128/5*t^12-29*t^11-24*t^10-7*t^9+128/5*t^8
 -7*t^7-24*t^6-29*t^5-128/5*t^4-87/5*t^3-8*t^2-t

X:= t^3*((1-t^(r-2)) div (1-t)) * ((1-t^r) div (1-t));
Y:= -t^(r+1);
A:=6*(r^2*Y+r*X)/&*[1-t^i: i in [1,1,r,r]] - (r-3)*(t+4*t^2+t^3)/(1-t)^4;