What is the formula (non-iterative form) for the product $\prod_{i=1}^{n}((2i-1)k+1)$

Question

I am searching for a compact (non-iterative) expression/formula for the product: $\prod_{i=1}^{n}((2i-1)k+1)$. The variable k needs to be kept.

Answer 1

Also, we could use the Gamma function for this. $$ \prod _{i=1}^{n} \big(\left( 2\,i-1 \right) k+1\big)={{2}^{n}{k}^{n}\frac{\Gamma \left( {\frac {2\,nk+k+1}{2k}} \right)}{ \Gamma \left({\frac {k+1}{2k}} \right) }} $$

Answer 2

$$\prod_{i=1}^n((2i-1)k+1)=\sum_{d=0}^n\sum_{S\subseteq\{1,\ldots, n\}\\|S|=d}\prod_{i\in S}(2i-1)k^d$$