I am trying to find a closed form for the following expression:
$$ \prod_{k=1}^{l} \left( 2^kn + 2^k -1 \right), $$
where $n$ is a positive integer. I know that it is simple to find an expression for the similar product (without the "$-1$"):
$$ \prod_{k=1}^{l} \left( 2^kn + 2^k \right) = 2^{\frac{k(1+k)}{2}}(1+n)^k, $$
however I am unable to find such a closed form for the first product. Any help/pointers would be appreciated.
For what it is worth, here are the first few polynomials:
$$2n+1$$
$$8n^2+10n+3$$
$$64n^3+136n^2+94n+21$$
$$1024n^4+3136n^3+3544n^2+1746n+315$$
$$32768n^5+132096n^4+210624n^3+165736n^2+64206n+9765$$
The leading coefficient is of course $2$ raised to a triangular number, and the constant coefficient is the product of the powers of two minus one, which doesn't seem to have a closed-form. There is no reason that the other coefficients be any simpler.