Coefficient of x^n in Logarithm Expansion

If ${\log }_e\frac{1}{1+x+x^2+x^3}$ be expanded in a series of ascending powers of $x$, shew that the coefficient of $x^n$ is $-\frac{1}{n}$ if $n$ be odd, or of the form $4m+2$, and $\frac{3}{n}$ if $n$ be of the form $4m$.

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XVII. Exponential and Logarithmic Series, Examples XVII, No. 20