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.