Changes

F(P,A) &=  \sum_{i=1}^{P} \| \{\text{i distinct alleles observed in a set of P alleles}\} \| \\

F(P,A) &=  \sum_{i=1}^{P} \| \{\text{i distinct alleles observed in a set of P alleles}\} \| \\

           &=  \sum_{i=1}^{P-1} \| \{\text{i distinct alleles observed in a set of P alleles}\} \|  + \| \{\text{P distinct alleles observed in a set of P alleles}\} \| \\

           &=  \sum_{i=1}^{P-1} \| \{\text{i distinct alleles observed in a set of P alleles}\} \|  + \| \{\text{P distinct alleles observed in a set of P alleles}\} \| \\

           &=  \sum_{i=1}^{P-1} \| \{\text{no ways to choose i distinct alleles}\} \|\| \{\text{no ways to arrange i distinct alleles in a set of P alleles}\} \| + \| \{\text{P distinct alleles observed in a set of P alleles}\} \| \\

           &=  \sum_{i=1}^{P-1} \| \{\text{i distinct alleles observed in a set of P alleles}\} \| + \binom{A}{P}\\

          &=  \sum_{i=1}^{P-1} \| \{\text{no ways to choose i distinct alleles}\} \|\| \{\text{no ways to arrange i distinct alleles in a set of P alleles}\} \| +\binom{A}{P} \\

  &=  \sum_{i=1}^{P-1} \binom{A}{i}\binom{P-1}{i-1}  + \binom{A}{P}  

  &=  \sum_{i=1}^{P-1} \binom{A}{i}\binom{P-1}{i-1}  + \binom{A}{P}  

   \end{align}

   \end{align}