Power Sum Formula Grid

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Main doc

Key:

Column number j refers to sum of of degree j $\left(\displaystyle \sum_{k=1}^{n} k^j \right)$.

Row number i refers to the coefficient $a_i$.

All numbers accurate to 5 decimal places (unless the number has 10+ digits).

As an example, the 3rd column from the right reads:

$\begin{array}{c}\\ \frac{1}{3}\\ \frac{1}{2}\\ \frac{1}{6} \end{array}$

This means $a_3=\frac{1}{3}$, $a_2=\frac{1}{2}$, and $a_1=\frac{1}{6}$. Thus the closed form solution is $a_3n^3+a_2n^2+a_1n=\frac{1}{3}n^3+\frac{1}{2}n^2+\frac{1}{6}n$.

The right most column should be formatted like all the others, it is not colored differently for any mathematical purpose.

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10&\color{#D61F06}{0}&9.43E+33&\color{#D61F06}{0}&-1.09E+32&\color{#D61F06}{0}&1.34E+30&\color{#D61F06}{0}&-1.79E+28&\color{#D61F06}{0}&2.56E+26&\color{#D61F06}{0}&-3.96E+24&\color{#D61F06}{0}&6.65E+22&\color{#D61F06}{0}&-1.22E+21&\color{#D61F06}{0}&2.42E+19&\color{#D61F06}{0}&-5.30E+17&\color{#D61F06}{0}&1.27E+16&\color{#D61F06}{0}&-3.40E+14&\color{#D61F06}{0}&10065214632473&\color{#D61F06}{0}&-333914927986&\color{#D61F06}{0}&12483366789&\color{#D61F06}{0}&-529918193.4&\color{#D61F06}{0}&25764029.11&\color{#D61F06}{0}&-1448909.917&\color{#D61F06}{0}&95338.83333&\color{#D61F06}{0}&-7439.76667&\color{#D61F06}{0}&699.83333&\color{#D61F06}{0}&-81.03333&\color{#D61F06}{0}&11.91667&\color{#D61F06}{0}&-2.38333&\color{#D61F06}{0}&0.91667&0.5&0.1&\color{#D61F06}{0}&\color{#D61F06}{0}&\color{#D61F06}{0}&\color{#D61F06}{0}&\color{#D61F06}{0}&\color{#D61F06}{0}&\color{#D61F06}{0}&\color{#D61F06}{0}&0\\ 9&-1.43E+35&\color{#D61F06}{0}&1.60E+33&\color{#D61F06}{0}&-1.91E+31&\color{#D61F06}{0}&2.45E+29&\color{#D61F06}{0}&-3.37E+27&\color{#D61F06}{0}&5.02E+25&\color{#D61F06}{0}&-8.09E+23&\color{#D61F06}{0}&1.42E+22&\color{#D61F06}{0}&-2.70E+20&\color{#D61F06}{0}&5.63E+18&\color{#D61F06}{0}&-1.29E+17&\color{#D61F06}{0}&3.27E+15&\color{#D61F06}{0}&-91783750390075&\color{#D61F06}{0}&2875775609278&\color{#D61F06}{0}&-101186341814&\color{#D61F06}{0}&4026892512&\color{#D61F06}{0}&-182730411.5&\color{#D61F06}{0}&9542233.003&\color{#D61F06}{0}&-579563.9667&\color{#D61F06}{0}&41451.66667&\color{#D61F06}{0}&-3542.74603&\color{#D61F06}{0}&368.33333&\color{#D61F06}{0}&-47.66667&\color{#D61F06}{0}&7.94444&\color{#D61F06}{0}&-1.83333&\color{#D61F06}{0}&0.83333&0.5&0.11111&\color{#D61F06}{0}&\color{#D61F06}{0}&\color{#D61F06}{0}&\color{#D61F06}{0}&\color{#D61F06}{0}&\color{#D61F06}{0}&\color{#D61F06}{0}&0\\ 8&\color{#D61F06}{0}&-2.15E+34&\color{#D61F06}{0}&2.48E+32&\color{#D61F06}{0}&-3.07E+30&\color{#D61F06}{0}&4.08E+28&\color{#D61F06}{0}&-5.84E+26&\color{#D61F06}{0}&9.04E+24&\color{#D61F06}{0}&-1.52E+23&\color{#D61F06}{0}&2.77E+21&\color{#D61F06}{0}&-5.52E+19&\color{#D61F06}{0}&1.21E+18&\color{#D61F06}{0}&-2.91E+16&\color{#D61F06}{0}&7.74E+14&\color{#D61F06}{0}&-22945937597519&\color{#D61F06}{0}&761234720103&\color{#D61F06}{0}&-28458658635&\color{#D61F06}{0}&1208067754&\color{#D61F06}{0}&-58734775.13&\color{#D61F06}{0}&3303080.655&\color{#D61F06}{0}&-217336.4875&\color{#D61F06}{0}&16957.5&\color{#D61F06}{0}&-1594.23571&\color{#D61F06}{0}&184.16667&\color{#D61F06}{0}&-26.8125&\color{#D61F06}{0}&5.10714&\color{#D61F06}{0}&-1.375&\color{#D61F06}{0}&0.75&0.5&0.125&\color{#D61F06}{0}&\color{#D61F06}{0}&\color{#D61F06}{0}&\color{#D61F06}{0}&\color{#D61F06}{0}&\color{#D61F06}{0}&0\\ 7&2.61E+35&\color{#D61F06}{0}&-2.91E+33&\color{#D61F06}{0}&3.48E+31&\color{#D61F06}{0}&-4.46E+29&\color{#D61F06}{0}&6.15E+27&\color{#D61F06}{0}&-9.16E+25&\color{#D61F06}{0}&1.48E+24&\color{#D61F06}{0}&-2.58E+22&\color{#D61F06}{0}&4.92E+20&\color{#D61F06}{0}&-1.03E+19&\color{#D61F06}{0}&2.36E+17&\color{#D61F06}{0}&-5.96E+15&\color{#D61F06}{0}&1.67E+14&\color{#D61F06}{0}&-5244785736576&\color{#D61F06}{0}&184541750328&\color{#D61F06}{0}&-7344169970&\color{#D61F06}{0}&333260070&\color{#D61F06}{0}&-17402896.33&\color{#D61F06}{0}&1056985.81&\color{#D61F06}{0}&-75595.3&\color{#D61F06}{0}&6460&\color{#D61F06}{0}&-671.25714&\color{#D61F06}{0}&86.66667&\color{#D61F06}{0}&-14.3&\color{#D61F06}{0}&3.14286&\color{#D61F06}{0}&-1&\color{#D61F06}{0}&0.66667&0.5&0.14286&\color{#D61F06}{0}&\color{#D61F06}{0}&\color{#D61F06}{0}&\color{#D61F06}{0}&\color{#D61F06}{0}&0\\ 6&\color{#D61F06}{0}&3.05E+34&\color{#D61F06}{0}&-3.52E+32&\color{#D61F06}{0}&4.35E+30&\color{#D61F06}{0}&-5.78E+28&\color{#D61F06}{0}&8.28E+26&\color{#D61F06}{0}&-1.28E+25&\color{#D61F06}{0}&2.15E+23&\color{#D61F06}{0}&-3.93E+21&\color{#D61F06}{0}&7.83E+19&\color{#D61F06}{0}&-1.71E+18&\color{#D61F06}{0}&4.12E+16&\color{#D61F06}{0}&-1.10E+15&\color{#D61F06}{0}&32548733709352&\color{#D61F06}{0}&-1079808828119&\color{#D61F06}{0}&40368507884&\color{#D61F06}{0}&-1713639660&\color{#D61F06}{0}&83315017.5&\color{#D61F06}{0}&-4685395.167&\color{#D61F06}{0}&308287.5278&\color{#D61F06}{0}&-24053.05&\color{#D61F06}{0}&2261&\color{#D61F06}{0}&-261.04444&\color{#D61F06}{0}&37.91667&\color{#D61F06}{0}&-7.15&\color{#D61F06}{0}&1.83333&\color{#D61F06}{0}&-0.7&\color{#D61F06}{0}&0.58333&0.5&0.16667&\color{#D61F06}{0}&\color{#D61F06}{0}&\color{#D61F06}{0}&\color{#D61F06}{0}&0\\ 5&-2.78E+35&\color{#D61F06}{0}&3.10E+33&\color{#D61F06}{0}&-3.70E+31&\color{#D61F06}{0}&4.74E+29&\color{#D61F06}{0}&-6.54E+27&\color{#D61F06}{0}&9.74E+25&\color{#D61F06}{0}&-1.57E+24&\color{#D61F06}{0}&2.75E+22&\color{#D61F06}{0}&-5.24E+20&\color{#D61F06}{0}&1.09E+19&\color{#D61F06}{0}&-2.51E+17&\color{#D61F06}{0}&6.34E+15&\color{#D61F06}{0}&-1.78E+14&\color{#D61F06}{0}&5579782921603&\color{#D61F06}{0}&-196328877840&\color{#D61F06}{0}&7813259591&\color{#D61F06}{0}&-354546136.5&\color{#D61F06}{0}&18514448.33&\color{#D61F06}{0}&-1124494.84&\color{#D61F06}{0}&80422.83333&\color{#D61F06}{0}&?????&\color{#D61F06}{0}&714&\color{#D61F06}{0}&-92.13333&\color{#D61F06}{0}&15.16667&\color{#D61F06}{0}&-3.3&\color{#D61F06}{0}&1&\color{#D61F06}{0}&-0.46667&\color{#D61F06}{0}&0.5&0.5&0.2&\color{#D61F06}{0}&\color{#D61F06}{0}&\color{#D61F06}{0}&0\\ 4&\color{#D61F06}{0}&-2.32E+34&\color{#D61F06}{0}&2.67E+32&\color{#D61F06}{0}&-3.30E+30&\color{#D61F06}{0}&4.39E+28&\color{#D61F06}{0}&-6.29E+26&\color{#D61F06}{0}&9.74E+24&\color{#D61F06}{0}&-1.64E+23&\color{#D61F06}{0}&2.99E+21&\color{#D61F06}{0}&-5.95E+19&\color{#D61F06}{0}&1.30E+18&\color{#D61F06}{0}&-3.13E+16&\color{#D61F06}{0}&8.35E+14&\color{#D61F06}{0}&-24734071673904&\color{#D61F06}{0}&820556312000&\color{#D61F06}{0}&-30676387162&\color{#D61F06}{0}&1302209932&\color{#D61F06}{0}&-63311810.09&\color{#D61F06}{0}&3560470.833&\color{#D61F06}{0}&-234269.7583&\color{#D61F06}{0}&18277.91667&\color{#D61F06}{0}&-1718.075&\color{#D61F06}{0}&198.33333&\color{#D61F06}{0}&-28.79167&\color{#D61F06}{0}&5.41667&\color{#D61F06}{0}&-1.375&\color{#D61F06}{0}&0.5&\color{#D61F06}{0}&-0.29167&\color{#D61F06}{0}&0.41667&0.5&0.25&\color{#D61F06}{0}&\color{#D61F06}{0}&0\\ 3&1.41E+35&\color{#D61F06}{0}&-1.57E+33&\color{#D61F06}{0}&1.88E+31&\color{#D61F06}{0}&-2.40E+29&\color{#D61F06}{0}&3.32E+27&\color{#D61F06}{0}&-4.94E+25&\color{#D61F06}{0}&7.95E+23&\color{#D61F06}{0}&-1.39E+22&\color{#D61F06}{0}&2.65E+20&\color{#D61F06}{0}&-5.54E+18&\color{#D61F06}{0}&1.27E+17&\color{#D61F06}{0}&-3.21E+15&\color{#D61F06}{0}&90219075042845&\color{#D61F06}{0}&-2826751048446&\color{#D61F06}{0}&99461371152&\color{#D61F06}{0}&-3958243505&\color{#D61F06}{0}&179615163&\color{#D61F06}{0}&-9379527.421&\color{#D61F06}{0}&569675.3333&\color{#D61F06}{0}&-40742.56667&\color{#D61F06}{0}&3481.50794&\color{#D61F06}{0}&-361.7&\color{#D61F06}{0}&46.66667&\color{#D61F06}{0}&-7.67778&\color{#D61F06}{0}&1.66667&\color{#D61F06}{0}&-0.5&\color{#D61F06}{0}&0.22222&\color{#D61F06}{0}&-0.16667&\color{#D61F06}{0}&0.33333&0.5&0.33333&\color{#D61F06}{0}&0\\ 2&\color{#D61F06}{0}&7.04E+33&\color{#D61F06}{0}&-8.12E+31&\color{#D61F06}{0}&1.00E+30&\color{#D61F06}{0}&-1.34E+28&\color{#D61F06}{0}&1.91E+26&\color{#D61F06}{0}&-2.96E+24&\color{#D61F06}{0}&4.97E+22&\color{#D61F06}{0}&-9.08E+20&\color{#D61F06}{0}&1.81E+19&\color{#D61F06}{0}&-3.96E+17&\color{#D61F06}{0}&9.52E+15&\color{#D61F06}{0}&-2.54E+14&\color{#D61F06}{0}&7518256253570&\color{#D61F06}{0}&-249419210157&\color{#D61F06}{0}&9324503545&\color{#D61F06}{0}&-395824350.5&\color{#D61F06}{0}&19244481.75&\color{#D61F06}{0}&-1082253.164&\color{#D61F06}{0}&71209.41667&\color{#D61F06}{0}&-5555.80455&\color{#D61F06}{0}&522.22619&\color{#D61F06}{0}&-60.28333&\color{#D61F06}{0}&8.75&\color{#D61F06}{0}&-1.64524&\color{#D61F06}{0}&0.41667&\color{#D61F06}{0}&-0.15&\color{#D61F06}{0}&0.08333&\color{#D61F06}{0}&-0.08333&\color{#D61F06}{0}&0.25&0.5&0.5&0\\ 1&-2.14E+34&\color{#D61F06}{0}&2.39E+32&\color{#D61F06}{0}&-2.85E+30&\color{#D61F06}{0}&3.65E+28&\color{#D61F06}{0}&-5.04E+26&\color{#D61F06}{0}&7.50E+24&\color{#D61F06}{0}&-1.21E+23&\color{#D61F06}{0}&2.12E+21&\color{#D61F06}{0}&-4.03E+19&\color{#D61F06}{0}&8.42E+17&\color{#D61F06}{0}&-1.93E+16&\color{#D61F06}{0}&4.88E+14&\color{#D61F06}{0}&-13711655205088&\color{#D61F06}{0}&429614643061&\color{#D61F06}{0}&-15116315767&\color{#D61F06}{0}&601580873.9&\color{#D61F06}{0}&-27298231.07&\color{#D61F06}{0}&1425517.167&\color{#D61F06}{0}&-86580.25311&\color{#D61F06}{0}&6192.12319&\color{#D61F06}{0}&-529.12424&\color{#D61F06}{0}&54.97118&\color{#D61F06}{0}&-7.09216&\color{#D61F06}{0}&1.16667&\color{#D61F06}{0}&-0.25311&\color{#D61F06}{0}&0.07576&\color{#D61F06}{0}&-0.03333&\color{#D61F06}{0}&0.02381&\color{#D61F06}{0}&-0.03333&\color{#D61F06}{0}&0.16667&0.5&1 \end{array}$