KL divergence between binomial distribution and Poisson distribution

Question

Let $$P(X=k) = {n\choose k}p^k(1-p)^{n-k} \\ Q(Y=k) = \frac{\lambda^k\exp(-\lambda)}{k!}$$ I'm trying to find $D_\text{KL}(P||Q)$. I obtained $$D_\text{KL}(P||Q) = \sum_{k}P(k)\ln(\frac{P(k)}{Q(k)}) = \sum_{k=0}^{n}{n\choose k}p^k(1-p)^{n-k}\ln(\frac{{n\choose k}p^k(1-p)^{n-k}}{\frac{\lambda^k\exp(-\lambda)}{k!}}) = \\ np\ln(p/\lambda) - np\ln(1-p)+n\ln(1-p) + \lambda +\ln(n!) - \sum_{k=0}^{n}{n\choose k}p^k(1-p)^{n-k}\ln((n-k)!)$$ Is there any closed-form expression for the last summation? $$F(n,p) = \sum_{k=0}^{n}{n\choose k}p^k(1-p)^{n-k}\ln((n-k)!)$$