Approximate formulas for integrals of Gamma functions?

[YaroslavVB]

Suppose f(n,p)=integral(n!/(x! (n - x)!dx, for x from -1/2 to p)
where n>1, p<n+1/2

Are approximate formulas known for this kind of integral?
Empirically, f(n,n+1/2) seems to be close to 2^n

More generally, I'm looking for approximate formulas for integrals of n!/(x1!x2!...xn!) over nice sets, textbook suggestions are welcome

ListPlot[Table[2^n - NIntegrate[n!/(x! (n - x)!), {x, -1/2, n + 1/2}], {n, Range[30]}], PlotRange -> {{0, 30}, {0, .12}}]

 

[JJacquelin]

In attachment, the proof of the 2^n rough approximate.
In order to obtain a very accurate approximate, one can use series developments of the Gamma function. The range of integration have to be split in two :
A first range for low values of x and the series development of Gamma in a range close to 0.
A second range for large values of x and the asymptotic series development of Gamma.
Indeed, the calculus will be rather arduous and the series development will have to be limited to a very low number of terms.

 

[YaroslavVB]

Thanks! Derivation in the attachment, where is it from?

 

[JJacquelin]

where is it from?

From nowhere.