nmax = 10;
nfmax = 3;
zs = Table[z[i], {i, nfmax}];
lnmt1[nmax_, nfmax_] := Sum[Sum[z[i]^n, {i, nfmax}], {n, nmax}];
lnmt2[nmax_, zs__] := Sum[(1/n) (Sum[x^(k*(n/2))*f[k, y, zs, n],
{k, 1, nmax}]/(1 - x^(2*n))), {n, 1, nmax}];
temp[i_, y_, zs__] :=
Simplify[Solve[
SeriesCoefficient[
lnmt1[nmax, nfmax] - lnmt2[nmax, zs], {x, 0, i/2}] == 0,
f[i, y, zs, 1]][[1, 1, 2]]];
For[i = 1, i <= nmax, i++,
Print["coefficient of ", x^{i/2}, " = ",
Expand[Simplify[temp[i, y, Sequence @@ zs]]]];