Operator = (1 + x^3) D[#, {x, 2}] - 6 x # &;
n = 20;
Sumb = Sum[Subscript[alpha, k] x^k, {k, 0, n}];
Alpha = Table[Subscript[alpha, k], {k, 0, n}];
res = Operator[Sumb] /. {Subscript[alpha, 0] -> Subscript[a, 0],
Subscript[alpha, 1] -> Subscript[a, 1]};
coefs = CoefficientList[res, x];
equs = Thread[coefs == 0];

For[k = 1; Alphas = {}; equsk = equs[[1]]; subs = {},
    k <= Length[equs] - 1, k++,
    solalphak = Solve[equsk, Subscript[alpha, k + 1]];
    AppendTo[Alphas, Subscript[alpha, k + 1] /. solalphak][[1]];
    AppendTo[subs, solalphak];
    equsk = equs[[k + 1]] /. Flatten[subs]
]

Alphas0 = Flatten[Alphas];
series = Subscript[a, 0] + x Subscript[a, 1] +
Sum[Alphas0[[k]] x^(k + 1), {k, 1, Length[Alphas0] - 2}]