## Quantum field theory – Problem in the proof of Wick's theorem

In the proof of Wick's theorem (https://arxiv.org/abs/math/0406251) the author
for the set $$i_1, i_2, …, i_m$$ of indices define the polynomial
$$P (b) _ {i_1, … i_m} = ( partial_ {i_1} + sum A ^ {i_1i} b_i) times … times ( partial_ {i_m} + sum A ^ {i_mi} b_i) textbf {1}$$
where $$textbf {1}$$ is the function identically equal to 1,$$A$$ a bilinear form and $$partial_ {i_k}$$ is defined by $$partial_ {i_k} (b, Ab) = 2 sum A ^ {i_ki} b_i$$ .

How to prove that

$$P (0) _ {i_1, … i_m} = begin {cases} sum A ^ {j_1j_2} …, A ^ {j_ {m-1} j_m} & m = 2n \ 0, & m = 2n + 1 end {cases}$$

where the sum is on all the partitions $$(j_1, j_2),. . . , (jm – 1, j_m)$$ in pairs of the set
$$i_1, i_2 ,. . ., i_m$$ of the indices.