Number of ways to write $2n-1$ as $4x+2y+4xy+1$ where $x$ and $y$ are nonnegative integers

Question

I see a claim in OEIS/A1227 which state that

"Number of ways to write $2n-1$ as $4x+2y+4xy+1$ where $x$ and $y$ are nonnegative integers equals number of odd divisors of $n$."

I can't see how the representation number relate to the odd divisor function. Any comments and links is appreciated.