This is a modification for page 327 of the computations on pages 172-176 in order to find the relative invariants of weight s for the differential equations 
             y''y' + a[0,1] y''y + a[1,0] y'y' + a[1,1] y'y + a[2,1] yy = 0, 
when 2<= s <= 9.

The version number of Mathematica used to evaluate this notebook:

$VersionNumber

The main results are summarized for page 327 at the end of this notebook.

u[s_,p_,q_,r_]:=Floor[ (1/(p + q + r))*(s-Sum[(p + q + k)*e[s, p, q, k],{k, r + 1, s - p - q}]-Sum[(p + j + k)*e[s, p, j, k],{j, q + 1, 1},{k, 0, s - p - j}]-Sum[(i + j + k)*e[s, i, j, k],{i, p + 1, 2},{j, (i - 1)^2, 1},{k, 0, s - i - j}])]

nM[s_]:=Module[{counter, seq},counter = 0;seq= Apply[Sequence,Reverse[Flatten[Table[{e[s, i, j, k], 0, u[s, i, j, k]},{i, 0, 2},{j, (i - 1)^2, 1},{k, 0, s - i - j}],2]]];Do[If[(Sum[(i + j + k)*e[s, i, j, k],{i,0,2},{j,(1 - i)^2,1},{k, 0, s - i - j}])==s,(counter = counter + 1;monM[s, counter]=Product[D[w[i, j][z],{z, k}]^e[s, i, j, k],{i, 0, 2},{j, (i - 1)^2, 1},{k, 0, s - i - j}] ),counter = counter],Evaluate[seq]];counter]

printM[s_]:=Do[ Print["monM[", s, ", ", i, "] = ", 
              monM[s, i]],{i, 1, nM[s]}]

printM[1]

printM[2]

wPolynomial[s_]:=Sum[x[s, i]*monM[s, i],{i,1,nM[s]}]

sub[s_]:=Flatten[ Table[D[w[i, j][z],{z, k}]->D[c[i, j][z],{z, k}],{i, 0, 2},{j, (i - 1)^2, 1},{k, 0, s - i - j}] ]

printM[2]

cPoly[s_]:=wPolynomial[s] /. sub[s]

(*   m = 2; c[0, 0][z_]=1;c[j_,i_][z_]:=c[i, j][z] /; j > i;  *)

printM[2]

cPoly[2]

The command involving     cOneStar[i_,j_[[z_] :=       at this point has been removed.  In its place, the following four commands are copied and pasted from page 328 of the Manuscript.

cOneStar[0, 1][z_] := c[0, 1][z]  +  rho'[z]/rho[z]

cOneStar[1, 0][z_]:= c[1, 0][z]  +  2*rho'[z]/rho[z]

cOneStar[1, 1][z_] := ( c[1, 1][z] + 2*c[0, 1][z]*rho'[z]/rho[z] 
                         + 2*c[1, 0][z]*rho'[z]/rho[z] 
                         + 2*(rho'[z]/rho[z])^2 + rho''[z]/rho[z] )

cOneStar[2, 1][z_] := ( c[2, 1][z] + c[1, 1][z]*rho'[z]/rho[z]
        + c[1, 0][z]*(rho'[z]/rho[z])^2 + c[0, 1][z]*rho''[z]/rho[z]
        + rho'[z]*rho''[z]/rho[z]^2 )

sub1[s_]:= Flatten[ Table[D[w[i, j][z],{z, k}]->D[cOneStar[i, j][z],{z, k}],{i, 0, 2},{j, (i - 1)^2, 1},{k, 0, s - i - j} ] ]

cOneStarPoly[s_]:=wPolynomial[s] /. sub1[s]

alpha[0][j_,zet_]=1;g[zet_]=f''[zet]/f'[zet];

alpha[1][j_,zet_]= g[zet]*Sum[-k,{k, 1, j}];

alpha[2][j_,zet_]=(D[g[zet], zet]*Sum[-k*(k + 1)/2,{k, 1, j}]+g[zet]^2*Sum[(1 + k)^2*k/2,{k, 1, j}] )//Simplify;

The command involving     cTwoStar[i_,j_[[zet_] :=       at this point has been removed.  In its place, the following four commands are copied and pasted from page 328 of the Manuscript.   
(The three preceding commands  for    alpha[i][j_,zet_] :=   are no longer needed.)

cTwoStar[0, 1][zet_]:= f'[zet]*c[0, 1][f[zet]]

cTwoStar[1, 0][zet_]:= f'[zet]*c[1, 0][f[zet]]  - f''[zet]/f'[
              zet]

cTwoStar[1, 1][zet_] := ( f'[zet]^2*c[1, 1][f[zet]] 
                      - f''[zet]*c[0, 1][f[zet]] )

cTwoStar[2, 1][zet_] := f'[zet]^3*c[2, 1][f[zet]]

sub2[s_]:=Flatten[ Table[D[w[i, j][z],{z, k}]->D[cTwoStar[i, j][zet],{zet, k}],{i, 0, 2},{j, (i - 1)^2, 1},{k, 0, s - i - j} ] ]

cTwoStarPoly[s_]:=wPolynomial[s] /. sub2[s]

firstCondition[s_]:=firstCondition[s] = Expand[rho[z]^s*(cOneStarPoly[s] - cPoly[s])];

secondCondition[s_]:=secondCondition[s] = Expand[f'[zet]^s*(cTwoStarPoly[s]-f'[zet]^s*(cPoly[s] /. z -> f[zet]))];

v[s_,p_]:=Floor[(1/p)*(s-Sum[(i + j + k)*e[s, i, j, k],{i, 0, 2},{j, (i - 1)^2, 1},{k, 0, s - i - j}]-Sum[t*h[s, t],{t, p + 1, s}])]

nA[s_]:=Module[{counter, seq1, seq2},counter = 0;seq1=Apply[Sequence,Reverse[Flatten[Table[{e[s, i, j, k], 0, u[s, i, j, k]},{i, 0, 2},{j, (i - 1)^2, 1},{k, 0, s - i - j}],2]]];seq2=Apply[Sequence,Reverse[Table[{h[s, t],0,v[s, t]},{t,1,s}]]];Do[If[(Sum[(i + j + k)*e[s, i, j, k],{i, 0, 2},{j, (i - 1)^2, 1},{k, 0, s - i - j}]+Sum[t*h[s, t],{t, 1, s}])==s,(counter = counter + 1;monA[s, counter]=Product[D[c[i, j][z],{z, k}]^e[s, i, j, k],{i, 0, 2},{j, (i - 1)^2, 1},{k, 0, s - i - j}]*rho[z]^(s-Sum[h[s, t],{t, 1, s}])*Product[D[rho[z],{z, t}]^h[s, t],{t, 1, s}]),counter = counter],Evaluate[seq1],Evaluate[seq2]];counter]

printA[s_]:=Do[Print["monA[", s, ", ", i, "] = ", 
              monA[s, i]],{i, 1, nA[s]}]

printA[2]

nB[s_]:=Module[{counter, seq1, seq2},counter = 0;seq1=Apply[Sequence,Reverse[Flatten[Table[{e[s, i, j, k], 0, u[s, i, j, k]},{i, 0, 2},{j, (i - 1)^2, 1},{k, 0, s - i - j}],2]]];seq2=Apply[Sequence,Reverse[Table[{h[s, t],0,v[s, t]},{t, 1, s}]]];Do[If[(Sum[(i + j + k)*e[s, i, j, k],{i, 0, 2},{j, (i - 1)^2, 1},{k, 0, s - i - j}]+ Sum[t*h[s, t],{t, 1, s}])==s,(counter = counter + 1;monB[s, counter]=Product[(D[c[i, j][z],{z, k}]/.z -> f[zet] )^e[s, i, j, k],{i, 0, 2},{j, (i - 1)^2, 1},{k, 0, s - i - j}]*f'[zet]^(2  s-Sum[(t + 1)*h[s, t],{t, 1, s}])*Product[D[f[zet],{zet, t + 1}]^h[s, t],{t, 1, s}]),counter = counter],Evaluate[seq1],Evaluate[seq2]];counter]

printB[s_]:=Do[ Print["monB[", s, ", ", i, "] = ", 
              monB[s, i]],{i, 1, nB[s]}]

firstCoefficients[s_]:=Do[a[s, i] = Coefficient[firstCondition[s], monA[s, i]],{i, 1, nA[s]}]

firstZeroCheck[s_]:=Expand[firstCondition[s]-Sum[a[s, i]*monA[s, i],{i, 1, nA[s]}]]

secondCoefficients[s_]:=Do[ b[s, j] = Coefficient[secondCondition[s], monB[s, j]],{j, 1, nB[s]}]

secondZeroCheck[s_]:=Expand[secondCondition[s]-Sum[ b[s, j]*monB[s, j],{j, 1, nB[s]}]]

eqs[s_]:=Join[Table[a[s, i] == 0,{i, 1, nA[s]}],Table[b[s, j] == 0,{j, 1, nB[s]}]]

rules[s_]:=Flatten[Solve[eqs[s],Table[x[s, i],{i, 1, nM[s]}]]]

(*  Print["firstZeroCheck[", s, "] = ", 
            firstZeroCheck[s]];  *)

(* Print["secondZeroCheck[", s, "] = ", 
            secondZeroCheck[s]]; *)

invariants[s_]:=Module[{expr, counter},counter = 0;
		Print["   "];
		Print["   Computation of invariants[", 
              s, "]"];
		nM[s];Print["Number of Monic Monomials = ", nM[s]];
	nA[s];Print["Number of firstCondition Terms = ", 
                nA[s]];
  nB[s]; Print["Number of secondCondition Terms = ", 
                nB[s]];
		firstCondition[s];  firstCoefficients[s];
		Print["firstZeroCheck[", s, "] = ",  
              firstZeroCheck[s]];
		secondCondition[s]; secondCoefficients[s];
			Print["secondZeroCheck[", s, "] = ", 
              secondZeroCheck[s] ]; 
		eqs[s];    rules[s]; 
		expr=Expand[(Sum[ x[s, i]*monM[s, i],{i, 1, nM[s]}])/.rules[s]];Do[ If[
                    Length[Coefficient[expr, x[s, i]]] != 
                      0, (counter = counter + 1; 
                      Inv[s, counter] = Coefficient[expr, x[s, i]]; 
                      Print["Inv[", s, ", ", counter, "] = ", 
                        Inv[s, counter]] ),  counter = counter],{i, 1, nM[s]}]; n[s] = counter; Print["n[", s, "] = ",  n[s] ]   ]

Date[]

Do[ (  invariants[i];  Print[ Date[] ] ),  {i, 2, 
        9} ]

Date[]

x[2, 1][
        z_]  =  Rel[2, 1][
          z]  +  w[0, 1][z]*w[1, 1][
            z] - w[0, 1][z]^2*w[1, 0][z]

x[1, 1][z_]  =  Rel[1, 1][z]  +  w[0, 1]'[z]  + 
        2*w[0, 1][z]*w[1, 0][z] - w[0, 1][z]^2

x[1, 0][z_]  =  Semi[1, 0][z] + 2*w[0, 1][z]

Inv[2, 1]

A[2, 1]  =  
      Expand[ Inv[2, 
            1]  /.   w[1, 1][z] -> x[1, 1][z] ]

\(reduceInv[n_, i_]\  := \ 
      Module[{}, \ \ \ 
		\ \ B[n, i, 0]\  = \ Inv[n, i]; 
		\ \ Do[\ 
          B[n, i, k]\  = \ 
            Expand[\ B[n, i, 
                  k - 1]\  /. \ 
					\ \ \ \ \ D[\ \(w[2, 1]\)[z], {z, 
                      7 - k}] -> 
                  D[\ \(x[2, 1]\)[z], \ {z, 7 - k}\ ]\ ], \ {k, 1, 
            7}]; 
		\ Do[\ 
          B[n, i, 7 + k]\  = \ 
            Expand[\ B[n, i, 
                  7 + k - 
                    1]\  /. 
						\ \ \ D[\ \(w[1, 1]\)[z], \ {z, 
                      9 - k}] -> D[\ \(x[1, 1]\)[z], \ {z, 9 - k}]\ ], \ {k, 
            1, 9}]; 
		\ Do[\ 
          B[n, i, 16 + k]\  = \ 
            Expand[\ B[n, i, 
                  16 + k - 
                    1]\  /. 
						\ \ \ D[\ \(w[1, 0]\)[z], {z, 
                      13 - k}] -> 
                  D[\ \(x[1, 0]\)[z], \ {z, 13 - k}]\ ], \ {k, 1, 
            13}]; 
\ \ \ \ \ G[n, i]\  = \ 
          B[n, i, 29]; 
Print["G[", n, \ ", ", \ 
          i, \ "] = ", \ G[n, i]]\ ]\)

Do[ reduceInv[s, i] ,   {s, 2, 9},  {i, 1, n[s]} ]

Expand[ G[2, 1]  -  Rel[1, 1][z] ]

Expand[ 
      G[3, 1]  -  ( 
          D[G[2, 1], z]  + 2  Semi[1, 0][z]*G[2, 1] ) ]

Expand[ G[3, 2]  -  Rel[2, 1][z] ]

Expand[ G[4, 1]  -  G[2, 1]^2 ]

Expand[ 
      G[4, 2]  -  ( 
          D[G[3, 1], z]  +  3  Semi[1, 0][z]*G[3, 1] ) ]

Expand[ 
      G[4, 3]  -  ( 
          D[G[3, 2], z]  +  3  Semi[1, 0][z]*G[3, 2] ) ]

Expand[ G[5, 1]  -  G[2, 1]*G[3, 1] ]

Expand[ 
      G[5, 2]  -  ( 
          D[G[4, 2], z]  +  4  Semi[1, 0][z]*G[4, 2] ) ]

Expand[ G[5, 3]  -  G[2, 1]*G[3, 2] ]

Expand[ 
      G[5, 4]  -  ( 
          D[G[4, 3], z]  +  4  Semi[1, 0][z]*G[4, 3] ) ]

Expand[ G[6, 1]  -  G[2, 1]^3 ]

Expand[ G[6, 2]  -  G[3, 1]^2 ]

Expand[ G[6, 3]  -  G[2, 1]*G[4, 2] ]

Expand[ 
      G[6, 4]  -  ( 
          D[G[5, 2], z]  +  5  Semi[1, 0][z]*G[5, 2] ) ]

Expand[ G[6, 5]  -  G[3, 1]*G[3, 2] ]

Expand[ G[6, 6]  -  G[3, 2]^2 ]

Expand[ G[6, 7]  -  G[2, 1]*G[4, 3] ]

Expand[ 
      G[6, 8]  -  ( 
          D[G[5, 4], z]  +  5  Semi[1, 0][z]*G[5, 4] ) ]

Expand[ G[7, 1]  -  G[2, 1]^2*G[3, 1] ]

Expand[ G[7, 2]  -  G[3, 1]*G[4, 2] ]

Expand[ G[7, 3]  -  G[2, 1]*G[5, 2] ]

Expand[ 
      G[7, 4]  -  ( 
          D[G[6, 4], z]  +  6  Semi[1, 0][z]*G[6, 4] ) ]

Expand[ G[7, 5]  -  G[2, 1]^2*G[3, 2] ]

Expand[ G[7, 6]  -  G[3, 2]*G[4, 2] ]

Expand[ G[7, 7]  -  G[3, 1]*G[4, 3] ]

Expand[ G[7, 8]  -  G[3, 2]*G[4, 3] ]

Expand[ G[7, 9]  -  G[2, 1]*G[5, 4] ]

Expand[ 
      G[7, 10]  -  ( 
          D[G[6, 8], z]  +  6  Semi[1, 0][z]*G[6, 8] ) ]

Expand[ G[8, 1]  -  G[2, 1]^4 ]

Expand[ G[8, 2]  -  G[2, 1]*G[3, 1]^2 ]

Expand[ G[8, 3]  -  G[2, 1]^2*G[4, 2] ]

Expand[ G[8, 4]  -  G[4, 2]^2 ]

Expand[ G[8, 5]  -  G[3, 1]*G[5, 2] ]

Expand[ G[8, 6]  -  G[2, 1]*G[6, 4] ]

Expand[ 
      G[8, 7]  -  ( 
          D[G[7, 4], z]  +  7  Semi[1, 0][z]*G[7, 4] ) ]

Expand[ G[8, 8]  -  G[2, 1]*G[3, 1]*G[3, 2] ]

Expand[ G[8, 9]  -  G[3, 2]*G[5, 2] ]

Expand[ G[8, 10]  -  G[2, 1]*G[3, 2]^2 ]

Expand[ G[8, 11]  -  G[2, 1]^2*G[4, 3] ]

Expand[ G[8, 12]  -  G[4, 2]*G[4, 3] ]

Expand[ G[8, 13]  -  G[4, 3]^2 ]

Expand[ G[8, 14]  -  G[3, 1]*G[5, 4] ]

Expand[ G[8, 15]  -  G[3, 2]*G[5, 4] ]

Expand[ G[8, 16]  -  G[2, 1]*G[6, 8] ]

Expand[ 
      G[8, 17]  -  ( 
          D[G[7, 10], z]  +  
            7  Semi[1, 0][z]*G[7, 10] ) ]

Expand[ G[9, 1]  -  G[2, 1]^3*G[3, 1] ]

Expand[ G[9, 2]  -  G[3, 1]^3 ]

Expand[ G[9, 3]  -  G[2, 1]*G[3, 1]*G[4, 2] ]

Expand[ G[9, 4]  -  G[2, 1]^2*G[5, 2] ]

Expand[ G[9, 5]  -  G[4, 2]*G[5, 2] ]

Expand[ G[9, 6]  -  G[3, 1]*G[6, 4] ]

Expand[ G[9, 7]  -  G[2, 1]*G[7, 4] ]

Expand[ 
      G[9, 8]  -  ( 
          D[G[8, 7], z]  +  8  Semi[1, 0][z]*G[8, 7] ) ]

Expand[ G[9, 9]  -  G[2, 1]^3*G[3, 2] ]

Expand[ G[9, 10]  -  G[3, 1]^2*G[3, 2] ]

Expand[ G[9, 11]  -  G[2, 1]*G[3, 2]*G[4, 2] ]

Expand[ G[9, 12]  -  G[3, 2]*G[6, 4] ]

Expand[ G[9, 13]  -  G[3, 1]*G[3, 2]^2 ]

Expand[ G[9, 14]  -  G[3, 2]^3 ]

Expand[ G[9, 15]  -  G[2, 1]*G[3, 1]*G[4, 3] ]

Expand[ G[9, 16]  -  G[4, 3]*G[5, 2] ]

Expand[ G[9, 17]  -  G[2, 1]*G[3, 2]*G[4, 3] ]

Expand[ G[9, 18]  -  G[2, 1]^2*G[5, 4] ]

Expand[ G[9, 19]  -  G[4, 2]*G[5, 4] ]

Expand[ G[9, 20]  -  G[4, 3]*G[5, 4] ]

Expand[ G[9, 21]  -  G[3, 1]*G[6, 8] ]

Expand[ G[9, 22]  -  G[3, 2]*G[6, 8] ]

Expand[ G[9, 23]  -  G[2, 1]*G[7, 10] ]

Expand[ 
      G[9, 24]  -  ( 
          D[G[8, 17], z]  +  
            8  Semi[1, 0][z]*G[8, 17] ) ]

We have also evaluated    invariants[10]    and found that it gives  n(10) = 36 .