I am trying to get the pattern of the Faa di Bruno's formula of the chain rule for higher derivatives. The only thing which I don't understand is how to get the coefficients of the various terms.
For example, let $y=g(x)$. Now $(f\circ g)''''(x)=f''''(y)y'^4+6f'''(y)y''y'^2+3f''(y)y''^2+4f''(y)y'''y'+f'(y)g''''$.
Here we note that that each term corresponds with a partition of $4$. The first term has four $y'$ terms in it and so it corresponds with $1+1+1+1$ (this sum has four terms and hence fourth derivative of $f(y)$ is there). The second term has a $y''$ and two $y'$ terms in it and so corresponds with $2+1+1$ (this sum has 3 terms and hence third derivative of $f(y)$ is there) and so on. But how can the pattern of the coefficients be expressed now?
Thanks