How many non-zero coefficients there are in Zhegalkin polynomial which is equal to: $$x_1 \lor x_2 \lor x_3 \lor x_4 \lor ... \lor x_n$$
Here under Zhegalkin polynomial following is understood:
for all sets S, such as $\oplus_{S \subseteq \{1,...,n\}}: a_S \land_{i \in S} x_i$, where $a_S \in \{0,1\}$ a_S - is mentioned coefficient here.
P.S: I think here a may use De Morgan law to transform all that disjunctions listed above into negative conjunctions as follows:
$$\neg x_1 \land \neg x_2 \land \neg x_3 \land \neg x_4 \land ... \land \neg x_n$$ to make it simpler to build up Zhegalkin polynomial itself, however can't find appropriate next step.
Looks like you are on a right way, but try to rewrite given expression as: $1\oplus(x_1\oplus1)\oplus(x_2\oplus1)\oplus...\oplus(x_n\oplus1)$, then simplify it by multiplying all the indices and get $2^{n-1}$ indices totally, after multiplying $1\oplus1$.