Based on the comments on OEIS-A002387:

$a_{n}$ = 1, 2, 4, 11, 31, 83, 227, 616,...

it is likely, that the sequence $a_{n}$ coincides with $[ e^{n-\gamma} ]$ , where $\gamma$ is the Euler-Mascheroni constant and $[\cdot]$ is the rounding function (remark made by Dean Hickerson).

My Question: Is there a formal proof, that OEIS-A002387 is $[ e^{n-\gamma} ]$ ?