alpha := (n,k) ->
-1/3 * exp(Pi*I/3+2*Pi*I*k/3) * (Pi*I/3 + 2*Pi*I*k/3)^n;

Q :=
proc(n)
option remember;
    local res;

    if n = 0 then return 2/9*sqrt(3)*Pi fi;

    res :=
    -1/(n+1)*add(alpha(n+1,k), k=0..2)
    -1/(n+1)*add(binomial(n+1, p)*(2*Pi*I)^(n-p)*Q(p),
                 p=0..n-1);

    simplify(res);
end;

VERIF := n -> int((log(x))^n/(x^3+1), x=0..infinity);