the following inequality is true, but I can't prove it The inequality $$\sum_{k=1}^{2d}\left(1-\frac{1}{2d+2-k}\right)\frac{d^k}{k!}>e^d\left(1-\frac{1}{d}\right)$$ holds for all integer $d\geq 1$. I use computer to verify it for $d\leq 50$, and find it is true, but I can't prove it.  Thanks for your answer.

Sorry I didn't check this sooner: the problem was cross-posted at
mathoverflow and I eventually was able to give a version of David Speyer's
analysis that makes it feasible to compute many more coefficient of the
asymptotic expansion (I reached the $d^{-13}$ term, and could go further),
and later to give error bounds good enough to prove the inequality for $d \geq 14$,
at which point the previous numerical computations complete the proof.
See 
https://mathoverflow.net/questions/133028/the-following-inequality-is-truebut-i-cant-prove-it/133123#133123
