Let $m_1,\dotsc,m_p$ be positive integers. Let $$ F(h_1,\dotsc,h_p) = \frac{1}{1-h_1-\dotsb-h_p} \prod_{i=1}^p \frac{(1-h_i)^{m_i}}{1-2h_i} $$ where the $h_i$ are variables. And let $$ G(z_1,\dotsc,z_p) = \prod_{i=1}^p \frac{\widehat{z}_i^{m_i} - z_i^{m_i}}{\widehat{z}_i - z_i} $$ where the $z_i$ are variables, and $\widehat{z}_i = (z_1 + \dotsb + z_p) - z_i$.
It is claimed that the coefficient of $h_1^{m_1-1} \dotsm h_p^{m_p-1}$ in $F$ is equal to the coefficient of $z_1^{m_1-1} \dotsm z_p^{m_p-1}$ in $G$. Is there an easy way to see this?
Here we are considering $F$ and $G$ as formal power series.
For example, with $(m_1,m_2,m_3,m_4,m_5)=(3,9,12,14,25)$, $h_1^2 h_2^8 h_3^{11} h_4^{13} h_5^{24}$ and $z_1^2 z_2^8 z_3^{11} z_4^{13} z_5^{24}$ each have coefficient $$ 1430462027777307645494624. $$
This is from https://arxiv.org/pdf/1708.00024.pdf, §9.5. The coefficients are equal because they each are equal to the Euclidean distance degree of a Segre variety. I'm curious if there's a more direct way to see the equality of coefficients.