Power Sum Formula Grid

(You may have to full screen this note for the table to render)

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:

\(\begin{array}\\ \frac{1}{3}\\ \frac{1}{2}\\ \frac{1}{6} \end{array}\)

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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20&\color{red}{0}&-1.35E+30&\color{red}{0}&1.56E+28&\color{red}{0}&-1.92E+26&\color{red}{0}&2.56E+24&\color{red}{0}&-3.66E+22&\color{red}{0}&5.67E+20&\color{red}{0}&-9.52E+18&\color{red}{0}&1.74E+17&\color{red}{0}&-3.47E+15&\color{red}{0}&75749069674134&\color{red}{0}&-1823448503317&\color{red}{0}&48574269646&\color{red}{0}&-1439682648&\color{red}{0}&47763870&\color{red}{0}&-1785975.875&\color{red}{0}&75871.25&\color{red}{0}&-3700.125&\color{red}{0}&210.83333&\color{red}{0}&-14.75833&\color{red}{0}&1.75&0.5&0.05&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&0\\ 19&4.10E+31&\color{red}{0}&-4.57E+29&\color{red}{0}&5.46E+27&\color{red}{0}&-6.99E+25&\color{red}{0}&9.65E+23&\color{red}{0}&-1.44E+22&\color{red}{0}&2.31E+20&\color{red}{0}&-4.05E+18&\color{red}{0}&7.72E+16&\color{red}{0}&-1.61E+15&\color{red}{0}&36950765694700&\color{red}{0}&-935101796573&\color{red}{0}&26256361971&\color{red}{0}&-822675799&\color{red}{0}&28947800&\color{red}{0}&-1152242.5&\color{red}{0}&52325&\color{red}{0}&-2740.83333&\color{red}{0}&168.66667&\color{red}{0}&-12.83333&\color{red}{0}&1.66667&0.5&0.05263&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&0\\ 18&\color{red}{0}&1.30E+31&\color{red}{0}&-1.50E+29&\color{red}{0}&1.85E+27&\color{red}{0}&-2.46E+25&\color{red}{0}&3.53E+23&\color{red}{0}&-5.46E+21&\color{red}{0}&9.16E+19&\color{red}{0}&-1.67E+18&\color{red}{0}&3.34E+16&\color{red}{0}&-7.29E+14&\color{red}{0}&17551613704982&\color{red}{0}&-467550898287&\color{red}{0}&13857524374&\color{red}{0}&-459730593.6&\color{red}{0}&17187756.25&\color{red}{0}&-729753.5833&\color{red}{0}&35506.25&\color{red}{0}&-2002.91667&\color{red}{0}&133.52778&\color{red}{0}&-11.08333&\color{red}{0}&1.58333&0.5&0.05556&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&0\\ 17&-3.55E+32&\color{red}{0}&3.96E+30&\color{red}{0}&-4.73E+28&\color{red}{0}&6.06E+26&\color{red}{0}&-8.36E+24&\color{red}{0}&1.24E+23&\color{red}{0}&-2.01E+21&\color{red}{0}&3.51E+19&\color{red}{0}&-6.69E+17&\color{red}{0}&1.40E+16&\color{red}{0}&-3.20E+14&\color{red}{0}&8100744786915&\color{red}{0}&-227457193761&\color{red}{0}&7126726821&\color{red}{0}&-250762141.9&\color{red}{0}&9979987.5&\color{red}{0}&-452950.5&\color{red}{0}&23670.83333&\color{red}{0}&-1442.1&\color{red}{0}&104.5&\color{red}{0}&-9.5&\color{red}{0}&1.5&0.5&0.05882&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&0\\ 16&\color{red}{0}&-1.01E+32&\color{red}{0}&1.16E+30&\color{red}{0}&-1.44E+28&\color{red}{0}&1.91E+26&\color{red}{0}&-2.73E+24&\color{red}{0}&4.23E+22&\color{red}{0}&-7.10E+20&\color{red}{0}&1.30E+19&\color{red}{0}&-2.59E+17&\color{red}{0}&5.65E+15&\color{red}{0}&-1.36E+14&\color{red}{0}&3624017404673&\color{red}{0}&-107410341498&\color{red}{0}&3563363410&\color{red}{0}&-133217387.9&\color{red}{0}&5655326.25&\color{red}{0}&-275005.6607&\color{red}{0}&15477.08333&\color{red}{0}&-1021.4875&\color{red}{0}&80.75&\color{red}{0}&-8.075&\color{red}{0}&1.41667&0.5&0.0625&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&0\\ 15&2.45E+33&\color{red}{0}&-2.73E+31&\color{red}{0}&3.26E+29&\color{red}{0}&-4.17E+27&\color{red}{0}&5.76E+25&\color{red}{0}&-8.57E+23&\color{red}{0}&1.38E+22&\color{red}{0}&-2.42E+20&\color{red}{0}&4.61E+18&\color{red}{0}&-9.62E+16&\color{red}{0}&2.21E+15&\color{red}{0}&-55812829431546&\color{red}{0}&1567142661480&\color{red}{0}&-49101870399&\color{red}{0}&1727691350&\color{red}{0}&-68757361.5&\color{red}{0}&3120180&\color{red}{0}&-162966.3175&\color{red}{0}&9905.33333&\color{red}{0}&-710.6&\color{red}{0}&61.52381&\color{red}{0}&-6.8&\color{red}{0}&1.33333&0.5&0.06667&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&0\\ 14&\color{red}{0}&6.11E+32&\color{red}{0}&-7.05E+30&\color{red}{0}&8.72E+28&\color{red}{0}&-1.16E+27&\color{red}{0}&1.66E+25&\color{red}{0}&-2.57E+23&\color{red}{0}&4.32E+21&\color{red}{0}&-7.88E+19&\color{red}{0}&1.57E+18&\color{red}{0}&-3.44E+16&\color{red}{0}&8.27E+14&\color{red}{0}&-22031380038768&\color{red}{0}&652976108950&\color{red}{0}&-21662589882&\color{red}{0}&809855320.5&\color{red}{0}&-34378680.75&\color{red}{0}&1671525&\color{red}{0}&-94019.0293&\color{red}{0}&6190.83333&\color{red}{0}&-484.5&\color{red}{0}&46.14286&\color{red}{0}&-5.66667&\color{red}{0}&1.25&0.5&0.07143&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&0\\ 13&-1.30E+34&\color{red}{0}&1.45E+32&\color{red}{0}&-1.73E+30&\color{red}{0}&2.22E+28&\color{red}{0}&-3.06E+26&\color{red}{0}&4.56E+24&\color{red}{0}&-7.35E+22&\color{red}{0}&1.29E+21&\color{red}{0}&-2.45E+19&\color{red}{0}&5.12E+17&\color{red}{0}&-1.17E+16&\color{red}{0}&2.97E+14&\color{red}{0}&-8336197852507&\color{red}{0}&261190443580&\color{red}{0}&-9190189647&\color{red}{0}&365741112.5&\color{red}{0}&-16596604.5&\color{red}{0}&866716.6667&\color{red}{0}&-52650.65641&\color{red}{0}&3768.33333&\color{red}{0}&-323&\color{red}{0}&34&\color{red}{0}&-4.66667&\color{red}{0}&1.16667&0.5&0.07692&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&0\\ 12&\color{red}{0}&-2.82E+33&\color{red}{0}&3.25E+31&\color{red}{0}&-4.02E+29&\color{red}{0}&5.35E+27&\color{red}{0}&-7.66E+25&\color{red}{0}&1.19E+24&\color{red}{0}&-1.99E+22&\color{red}{0}&3.63E+20&\color{red}{0}&-7.25E+18&\color{red}{0}&1.58E+17&\color{red}{0}&-3.81E+15&\color{red}{0}&1.02E+14&\color{red}{0}&-3010293668961&\color{red}{0}&99866934310&\color{red}{0}&-3733514544&\color{red}{0}&158487815.4&\color{red}{0}&-7705566.375&\color{red}{0}&433358.3333&\color{red}{0}&-28519.10556&\color{red}{0}&2226.74242&\color{red}{0}&-209.95&\color{red}{0}&24.55556&\color{red}{0}&-3.79167&\color{red}{0}&1.08333&0.5&0.08333&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&0\\ 11&5.14E+34&\color{red}{0}&-5.73E+32&\color{red}{0}&6.85E+30&\color{red}{0}&-8.78E+28&\color{red}{0}&1.21E+27&\color{red}{0}&-1.80E+25&\color{red}{0}&2.90E+23&\color{red}{0}&-5.08E+21&\color{red}{0}&9.69E+19&\color{red}{0}&-2.02E+18&\color{red}{0}&4.64E+16&\color{red}{0}&-1.17E+15&\color{red}{0}&32940702433548&\color{red}{0}&-1032100686501&\color{red}{0}&36315248840&\color{red}{0}&-1445231436&\color{red}{0}&65581165&\color{red}{0}&-3424696.167&\color{red}{0}&208012&\color{red}{0}&-14879.53333&\color{red}{0}&1272.42424&\color{red}{0}&-132.6&\color{red}{0}&17.33333&\color{red}{0}&-3.03333&\color{red}{0}&1&0.5&0.09091&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&0\\ 10&\color{red}{0}&9.43E+33&\color{red}{0}&-1.09E+32&\color{red}{0}&1.34E+30&\color{red}{0}&-1.79E+28&\color{red}{0}&2.56E+26&\color{red}{0}&-3.96E+24&\color{red}{0}&6.65E+22&\color{red}{0}&-1.22E+21&\color{red}{0}&2.42E+19&\color{red}{0}&-5.30E+17&\color{red}{0}&1.27E+16&\color{red}{0}&-3.40E+14&\color{red}{0}&10065214632473&\color{red}{0}&-333914927986&\color{red}{0}&12483366789&\color{red}{0}&-529918193.4&\color{red}{0}&25764029.11&\color{red}{0}&-1448909.917&\color{red}{0}&95338.83333&\color{red}{0}&-7439.76667&\color{red}{0}&699.83333&\color{red}{0}&-81.03333&\color{red}{0}&11.91667&\color{red}{0}&-2.38333&\color{red}{0}&0.91667&0.5&0.1&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&0\\ 9&-1.43E+35&\color{red}{0}&1.60E+33&\color{red}{0}&-1.91E+31&\color{red}{0}&2.45E+29&\color{red}{0}&-3.37E+27&\color{red}{0}&5.02E+25&\color{red}{0}&-8.09E+23&\color{red}{0}&1.42E+22&\color{red}{0}&-2.70E+20&\color{red}{0}&5.63E+18&\color{red}{0}&-1.29E+17&\color{red}{0}&3.27E+15&\color{red}{0}&-91783750390075&\color{red}{0}&2875775609278&\color{red}{0}&-101186341814&\color{red}{0}&4026892512&\color{red}{0}&-182730411.5&\color{red}{0}&9542233.003&\color{red}{0}&-579563.9667&\color{red}{0}&41451.66667&\color{red}{0}&-3542.74603&\color{red}{0}&368.33333&\color{red}{0}&-47.66667&\color{red}{0}&7.94444&\color{red}{0}&-1.83333&\color{red}{0}&0.83333&0.5&0.11111&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&0\\ 8&\color{red}{0}&-2.15E+34&\color{red}{0}&2.48E+32&\color{red}{0}&-3.07E+30&\color{red}{0}&4.08E+28&\color{red}{0}&-5.84E+26&\color{red}{0}&9.04E+24&\color{red}{0}&-1.52E+23&\color{red}{0}&2.77E+21&\color{red}{0}&-5.52E+19&\color{red}{0}&1.21E+18&\color{red}{0}&-2.91E+16&\color{red}{0}&7.74E+14&\color{red}{0}&-22945937597519&\color{red}{0}&761234720103&\color{red}{0}&-28458658635&\color{red}{0}&1208067754&\color{red}{0}&-58734775.13&\color{red}{0}&3303080.655&\color{red}{0}&-217336.4875&\color{red}{0}&16957.5&\color{red}{0}&-1594.23571&\color{red}{0}&184.16667&\color{red}{0}&-26.8125&\color{red}{0}&5.10714&\color{red}{0}&-1.375&\color{red}{0}&0.75&0.5&0.125&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&0\\ 7&2.61E+35&\color{red}{0}&-2.91E+33&\color{red}{0}&3.48E+31&\color{red}{0}&-4.46E+29&\color{red}{0}&6.15E+27&\color{red}{0}&-9.16E+25&\color{red}{0}&1.48E+24&\color{red}{0}&-2.58E+22&\color{red}{0}&4.92E+20&\color{red}{0}&-1.03E+19&\color{red}{0}&2.36E+17&\color{red}{0}&-5.96E+15&\color{red}{0}&1.67E+14&\color{red}{0}&-5244785736576&\color{red}{0}&184541750328&\color{red}{0}&-7344169970&\color{red}{0}&333260070&\color{red}{0}&-17402896.33&\color{red}{0}&1056985.81&\color{red}{0}&-75595.3&\color{red}{0}&6460&\color{red}{0}&-671.25714&\color{red}{0}&86.66667&\color{red}{0}&-14.3&\color{red}{0}&3.14286&\color{red}{0}&-1&\color{red}{0}&0.66667&0.5&0.14286&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&\color{red}{0}&0\\ 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3&1.41E+35&\color{red}{0}&-1.57E+33&\color{red}{0}&1.88E+31&\color{red}{0}&-2.40E+29&\color{red}{0}&3.32E+27&\color{red}{0}&-4.94E+25&\color{red}{0}&7.95E+23&\color{red}{0}&-1.39E+22&\color{red}{0}&2.65E+20&\color{red}{0}&-5.54E+18&\color{red}{0}&1.27E+17&\color{red}{0}&-3.21E+15&\color{red}{0}&90219075042845&\color{red}{0}&-2826751048446&\color{red}{0}&99461371152&\color{red}{0}&-3958243505&\color{red}{0}&179615163&\color{red}{0}&-9379527.421&\color{red}{0}&569675.3333&\color{red}{0}&-40742.56667&\color{red}{0}&3481.50794&\color{red}{0}&-361.7&\color{red}{0}&46.66667&\color{red}{0}&-7.67778&\color{red}{0}&1.66667&\color{red}{0}&-0.5&\color{red}{0}&0.22222&\color{red}{0}&-0.16667&\color{red}{0}&0.33333&0.5&0.33333&\color{red}{0}&0\\ 2&\color{red}{0}&7.04E+33&\color{red}{0}&-8.12E+31&\color{red}{0}&1.00E+30&\color{red}{0}&-1.34E+28&\color{red}{0}&1.91E+26&\color{red}{0}&-2.96E+24&\color{red}{0}&4.97E+22&\color{red}{0}&-9.08E+20&\color{red}{0}&1.81E+19&\color{red}{0}&-3.96E+17&\color{red}{0}&9.52E+15&\color{red}{0}&-2.54E+14&\color{red}{0}&7518256253570&\color{red}{0}&-249419210157&\color{red}{0}&9324503545&\color{red}{0}&-395824350.5&\color{red}{0}&19244481.75&\color{red}{0}&-1082253.164&\color{red}{0}&71209.41667&\color{red}{0}&-5555.80455&\color{red}{0}&522.22619&\color{red}{0}&-60.28333&\color{red}{0}&8.75&\color{red}{0}&-1.64524&\color{red}{0}&0.41667&\color{red}{0}&-0.15&\color{red}{0}&0.08333&\color{red}{0}&-0.08333&\color{red}{0}&0.25&0.5&0.5&0\\ 1&-2.14E+34&\color{red}{0}&2.39E+32&\color{red}{0}&-2.85E+30&\color{red}{0}&3.65E+28&\color{red}{0}&-5.04E+26&\color{red}{0}&7.50E+24&\color{red}{0}&-1.21E+23&\color{red}{0}&2.12E+21&\color{red}{0}&-4.03E+19&\color{red}{0}&8.42E+17&\color{red}{0}&-1.93E+16&\color{red}{0}&4.88E+14&\color{red}{0}&-13711655205088&\color{red}{0}&429614643061&\color{red}{0}&-15116315767&\color{red}{0}&601580873.9&\color{red}{0}&-27298231.07&\color{red}{0}&1425517.167&\color{red}{0}&-86580.25311&\color{red}{0}&6192.12319&\color{red}{0}&-529.12424&\color{red}{0}&54.97118&\color{red}{0}&-7.09216&\color{red}{0}&1.16667&\color{red}{0}&-0.25311&\color{red}{0}&0.07576&\color{red}{0}&-0.03333&\color{red}{0}&0.02381&\color{red}{0}&-0.03333&\color{red}{0}&0.16667&0.5&1 \end{array}\)