Let \([f_n(k)]\), \(k=0,1,\ldots,n\), be an arbitrary symmetric array of numbers in the sense that \(f_n(k)=f_n(n-k)\). A very special case would be the Pascal triangle with \(f_n(k)=\binom{n}{k}\). This paper considers expressions of the general form \(\displaystyle\sum_{k=0}^n k^pf_n(k) = \sum k^pf\), \(p\) an integer, the latter brief symbolism being used since we will retain a fixed \(n\) throughout and always sum on \(k\). Three theorems are proved. It is first shown that \[ 2\,\sum k^{2r+1}f = \sum_{j=0}^r (-1)^{r-j} n^{r+1-j} Q_j^r \sum k^{r+j}f, \] with \(Q_j^r = \binom{r}{j} + 2\binom{r}{j-1}\). The main result of interest is the expansion (brackets denote integer part) \[ 2^{[(r+s)/2]} \sum k^{2r+1}f = \sum_{j=0}^r A_j^r n^{2r-2j+1} \sum k^{2j}f. \] The coefficients \(A_j^r\) satisfy an involved recurrence relation and have the property that \[ A_r^{r+k} = \binom{2r+2k+1}{2k+1} 2^{[m/2]} A_0^k,\quad\text{for }r\ge 0, \] with \(m=r\) for odd \(k\) and \(m=r+1\) for even \(k\). Thus, knowledge of \(A_0^k\) determines the array. The first few values of \(A_0^k\) \((k=0,1,\ldots)\) are \(1,-1, 2, -17, 124, -2764, 43688,\ldots\). An example of the expansion is the formula \[ 4 \sum k^5 f = 2n^5 \sum f - 10n^3 \sum k^2 f + 10n \sum k^4f. \] Substitution of \(f_n(k)=\binom{n}{k}^p\), \(p\) an arbitrary real exponent, yields various binomial coefficient identities. It is remarked that \(A_j^r\) may be given explicitly in terms of the Bernoulli numbers, but this is not developed in the paper. \{Reviewer's remark: In a paper to appear, \textit{L. Carlitz} has found the actual expression for \(A_0^r\) in terms of the Bernoulli numbers, along with other valuable results.\} The third result in this paper is the expansion \[ C_r \sum k^{2r}f = \sum_{j=0}^r G_j^r n^{2r-2j-1} \sum k^{2j+1}f,\quad C_r = \frac{(2r+1)!}{r!2^r}, \] for which it develops that \[ \sum_{j=1}^r G_j^r A_1^j 2^{-[(j+3)/2]} = \begin{cases} 0 &\quad\text{if } i\ne r, \\ \frac{(2r+1)!}{r!2^r} &\quad\text{if } i = r.\end{cases} \] or, alternatively \[ \sum_{j=1}^r A_j^r G_1^j \frac{j!2^j}{(2j+1)!} = \begin{cases} 0 &\quad\text{if } i\ne r, \\ 2^{[(r+3)/2]} &\quad\text{if } i = r.\end{cases} \] It is also noted that no expansion of the form \[ \sum k^{2r}f = \sum_{j=0}^{r-1} E_j^r(n) \sum k^{2j+1}f \] can exist.