By the fundamental theorem of symmetric polynomials, $X_1,X_2,\cdots,X_n$ are polynomials of $ e_1,\cdots,e_n$ and $$ \mathbb{Z}[ e_1,\cdots,e_n]=\mathbb{Z}[X_1,X_2,\cdots,X_n]. $$ We define a polynomial $P_k(x_1,\cdots,x_k,y_1,\cdots,y_k)\in \mathbb{Z}_2[x_1,\cdots,x_k;y_1,\cdots,y_k]$ by letting $$ P(e_1,\cdots,e_k,e_1',\cdots,e_k')=\prod_{i,j=1}^k(1+X_i+X_j'). $$ Here $e_1,\cdots,e_k$ are the elementary symmetric polynomials of $X_1,\cdots,X_k$ and $e_1',\cdots,e_k'$ are the elementary symmetric polynomials of $X_1',\cdots,X_k'$.
Question: For integers $a\geq 1$, how to expand the following $$ f(x_1,\cdots,x_k)=\frac{P_k(x_1,\cdots,x_k;x_1,\cdots,x_k)}{(1+x_1+x_2+\cdots+x_k)^{a+k}} $$
into a Taylor series at zero ?