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I have difficulties to find an approximation formula (or bound from the below) for the following sum:

$$ \sum_{k=1}^n\left( \frac{1}{35}\right)^{k-1}(n-k)!\left(k-\frac 32\right)!. $$

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@David: Could you throw light on what you mean by $\left( k - \frac32\right)!$? –  user17762 Dec 27 '11 at 17:15
@Sivaram The close connection with the preceding question should give a strong hint :-). –  whuber Dec 27 '11 at 17:54
I've use representation of $\beta$ function in terms of $\Gamma$ function. Now I got the following sum and I have no idea how to approximate it. $\Gamma(k-0.5)=(k-3/2)!$ –  David Dec 27 '11 at 18:36

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