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Let
$$F (n,k)=frac {1}{2^n}sum_{i=0}^n binom{n}{i}(2i-n)^{2k},$$
where $n,k$ are non-negative integers.

By numerical tests the expression is an integer polynomial in $n $ of order $k $:
$$ F(n,0)=1;
F (n,1)=n;
F (n,2)=n (3n-2),$$
and so on.

Is there a simple general expression for the polynomial?

It is convenient to use the coefficient of operator $[z^n]$ to denote the coefficient of $z^n$ of a series. This way we can write for instance

begin{align*}
n![z^n]e^{jz}=j^ntag{1}
end{align*}

We obtain
begin{align*}
color{blue}{frac{1}{2^n}}&color{blue}{sum_{j=0}^nbinom{n}{j}(2j-n)^{2k}}\
&=frac{1}{2^n}sum_{j=0}^nbinom{n}{j}(2k)![z^{2k}]e^{(2j-n)z}tag{2}\
&=frac{(2k)!}{2^n}[z^{2k}]e^{-nz}sum_{j=0}^nbinom{n}{j}left(e^{2z}right)^jtag{3}\
&=frac{(2k)!}{2^n}[z^{2k}]e^{-nz}left(1+e^{2z}right)^ntag{4}\
&=(2k)![z^{2k}]left(frac{e^{z}+e^{-z}}{2}right)^ntag{5}\
&,,color{blue}{=(2k)![z^{2k}]left(cosh zright)^n}
end{align*}

We see OPs formula is essentially the coefficient of $z^{2k}$ of $left(cosh zright)^n$ which does not have a closed formula as far as I know.

Comment:

• In (2) we apply the coefficient of operator according to (1).




• In (3) use the linearity of the coefficient of operator.




• In (4) we apply the binomial theorem.




• In (5) we write the expression somewhat more conveniently.

I'll play with the cosh and
see if I get
anything other than
the original problem.

$begin{array}\
cosh^n(x)
&=frac1{2^n}(e^x+e^{-x})^n\
&=frac1{2^n}sum_{k=0}^n binom{n}{k}e^ke^{(n-k)x}\
&=frac1{2^n}e^{nx}sum_{k=0}^n binom{n}{k}e^{-2kx}\
&=frac1{2^n}sum_{i=0}^{infty} dfrac{(nx)^i}{i!}sum_{k=0}^n binom{n}{k}sum_{j=0}^{infty} dfrac{(-2kx)^j}{j!}\
&=frac1{2^n}sum_{k=0}^n binom{n}{k}sum_{i=0}^{infty} dfrac{(nx)^i}{i!}sum_{j=0}^{infty} dfrac{(-2kx)^j}{j!}\
&=frac1{2^n}sum_{k=0}^n binom{n}{k}sum_{m=0}^{infty}sum_{i=0}^{m} dfrac{(nx)^i}{i!} dfrac{(-2kx)^{m-i}}{(m-i)!}\
&=frac1{2^n}sum_{k=0}^n binom{n}{k}sum_{m=0}^{infty}x^msum_{i=0}^{m} dfrac{(n)^i}{i!} dfrac{(-2k)^{m-i}}{(m-i)!}\
&=frac1{2^n}sum_{m=0}^{infty}x^msum_{k=0}^n binom{n}{k}sum_{i=0}^{m} dfrac{(n)^i}{i!} dfrac{(-2k)^{m-i}}{(m-i)!}\
&=frac1{2^n}sum_{m=0}^{infty}x^msum_{i=0}^{m}dfrac{(-2)^{m-i}(n)^i}{(m-i)!i!}sum_{k=0}^n binom{n}{k} k^{m-i}\
&=frac1{2^n}sum_{m=0}^{infty}dfrac{x^m}{m!}sum_{i=0}^{m}binom{m}{i}(-2)^{m-i}n^isum_{k=0}^n binom{n}{k} k^{m-i}\
&=frac1{2^n}sum_{m=0}^{infty}dfrac{(-2)^mx^m}{m!}sum_{i=0}^{m}binom{m}{i}(-n/2)^isum_{k=0}^n binom{n}{k} k^{m-i}\
&=frac1{2^n}sum_{m=0}^{infty}dfrac{(-2)^mx^m}{m!}sum_{k=0}^n binom{n}{k}sum_{i=0}^{m}binom{m}{i}(-n/2)^i k^{m-i}\
&=frac1{2^n}sum_{m=0}^{infty}dfrac{(-2)^mx^m}{m!}sum_{k=0}^n binom{n}{k}(-n/2+k)^m\
&=frac1{2^n}sum_{m=0}^{infty}dfrac{x^m}{m!}sum_{k=0}^n binom{n}{k}(2k-n)^m\
end{array}
$

And this is the OP.

Oh well.

Let $omega $ be a $n$-dimensional vector with binary components $omega_i=pm1$ and $Omega_n $ be a set of all such vectors, the size of the set obviously being $2^n $. The sum of elements of a vector with $i$
positive and $n-i $ negative components is $2i-n $ and the number of such vectors is $binom {n}{i}$. Thus
$$
F (n,k)=frac {1}{2^n}sum_{omegainOmega_n } left (sum_{i=1}^nomega_i right)^{2k}
=frac {1}{2^n}sum_{omegainOmega_n } sum_{p_ige0}^{sum_i p_i=2k}binom {2k} {p_1,p_2,dots,p_n}prod_i omega_i^{p_i}
=sum_{p_ige0}^{sum_i p_i=2k}binom {2k} {p_1,p_2,dots,p_n}left (frac {1}{2^n}sum_{omegainOmega_n }prod_i omega_i^{p_i}right)
=sum_{p_ige0,;p_i,text {mod},2=0}^{sum_i p_i=2k}binom {2k} {p_1,p_2,dots,p_n}.
$$

To proceed further one splits the last sum into partial ones over terms with particular count $l$ of non-zero $p_i$ and ends up with:
$$
F (n,k)=sum_{l=1}^n T (k,l)frac{n!}{(n-l)!},tag {1}
$$
where $T (k,l)$ is the number of partitions of a set of size $2k$ into $l$ blocks of even size. It is the OEIS sequence A156289 with known close-form and recurrence expressions.

By numerical evidence the polynomial (1) can be expressed in the terms of usual powers as:
$$
F (n,k)=sum_{l=1}^n A (k,l)n^l,tag {2}
$$
with $A (k,l) $ being the OEIS sequence A318146. In other words $F(n,k) $ is in fact the so called Omega polynomial.