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.