On the Brousseau sums ${\sum }_{i=1}^n{i}^p{F}_i$

We start with new convolution formulas for $F_n-n^p$ involving only the binomial coefficients. Then, we use those to find direct formulas for the sums ${\sum }_{i=1}^n{i}^p{F}_{n-i}$ and ${\sum }_{i=1}^n{i}^p{F}_i$, and we show how our formulas connect to work in earlier papers by Ledin, Brousseau, Zeitlin, Adegoke, Shannon and Ollerton, and Kinlaw, Morris, and Thiagarajan.