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From my very old cookbook.

For integer values of $p$ $$A_p=\sum_{k=0}^n k^p\,\binom{n}{k}=2^{n-p}\,n\, B_p(n)$$ where the first polynomials are $$\left( \begin{array}{cc} p & B_p(n) \\ 1 & 1 \\ 2 & n+1 \\ 3 & n^2+3 n \\ 4 & n^3+6 n^2+3 n-2 \\ 5 & n^4+10 n^3+15 n^2-10 n \\ 6 & n^5+15 n^4+45 n^3-15 n^2-30 n+16 \\ 7 & n^6+21 n^5+105 n^4+35 n^3-210 n^2+112 n \\ 8 & n^7+28 n^6+210 n^5+280 n^4-735 n^3+28 n^2+588 n-272 \\ 9 & n^8+36 n^7+378 n^6+1008 n^5-1575 n^4-2436 n^3+5292 n^2-2448 n \\ \end{array} \right)$$

Tha $A_p$ are in fact the expansion of generalized hypergeometric functions. For example $$A_4=n \, _4F_3(2,2,2,1-n;1,1,1;-1)$$ $$A_5=n \, _5F_4(2,2,2,2,1-n;1,1,1,1;-1)$$ $$A_6=n \, _6F_5(2,2,2,2,2,1-n;1,1,1,1,1;-1)$$ The pattern is clear.

Asymptotically,

$$A_p=2^{n-p}\,n^p \left(1+\frac {p(p+1)}{2n}+ O\left(\frac{1}{n^2}\right) \right)$$ which seems to works decently for non iteger values of $p$.

Using $p=\frac 12$ and $n=5^m$, some numbers

$$\left( \begin{array}{cc} m & \frac {\text{approximate}}{\text{exact}} \\ 1 & 1.0389688 \\ 2 & 1.0052319 \\ 3 & 1.0010086 \\ 4 & 1.0002003 \\ 5 & 1.0000400 \\ 6 & 1.0000080 \\ 7 & 1.0000016 \\ \end{array} \right)$$