I am trying to prove that these two integrals are equal $$ \int_0^\infty\int_0^\infty\dots\int_0^\infty \exp(-x z_1-z_1z_2-z_2z_3-\dots-z_{n-1}z_{n}-z_n) x^{c_1-1} z_1^{c_1+c_2-1} z_2^{c_2+c_3-1} \dots z_{n}^{c_n+c_{n+1}-1} dz_1 dz_2 \dots dz_{n} $$ $$ \int_0^\infty\int_0^\infty\dots\int_0^\infty \exp(-x ^{w_n} z_1-z_1z_2-z_2z_3-\dots-z_{n-1}z_{n}-z_n) x^{{w_n}\,c_{n+1}-1} z_n^{c_1+c_2-1} z_{n-1}^{c_2+c_3-1} \dots z_{1}^{c_n+c_{n+1}-1} dz_1 dz_2 \dots dz_{n} $$ where $x>0,c_n>0,\forall n$ and $$w_n = \begin{cases} 1 & \text{even}\,\,n \\ -1 & \text{odd}\,\,n \end{cases}.$$
Using Mathematica, I was able to prove up to $n=5$, but is this valid for all $n$?
