Let $\mathbb{F}$ be a field of characteristic $2$, $n$ a positive integer and $f_n:\bigoplus\limits_{i=1}^n\mathbb{F}\sigma_{i}\mapsto \bigoplus\limits_{i,j=1,i<j}^n\mathbb{F}\sigma_{i,j}$ where we identify $\sigma_{i,j}=\sigma_{j,i}~\forall~i,j\in\lbrace 1,\dots,n\rbrace$, which is defined as: $$f_n(\sigma_i)=\sum_{j<i}\binom{i}{j}\sigma_{i,j}+\sum_{i<j}\binom{n-i+1}{j}\sigma_{i,j}~\forall~i=1,\dots,n$$ I want to compute $\dim_{\mathbb{F}}(\ker(f_n))$. Since we are in characteristic $2$ we would need to know when the combinatorial numbers are even or odd, for which there are theorems like Kummer's theorem, or Lucas' theorem. Sierpinski-Pascal-Triangle give us a geometric description of the parity of the combinatorial numbers, however I don't have ideas on how to use these facts to solve the problem.

Thanks for your help.