$\displaystyle{\sum_{k=1}^{n}k^{1}=\frac{1}{2}\left(n^{2}+n\right)=\frac{1}{2}n\left(n+1\right)}$
$\displaystyle{\sum_{k=1}^{n}k^{2}=\frac{1}{6}\left(2n^{3}+3n^{2}+n\right)=\frac{1}{6}n\left(n+1\right)\left(2n+1\right)}$
$\displaystyle{\sum_{k=1}^{n}k^{3}=\frac{1}{4}\left(n^{4}+2n^{3}+n^{2}\right)=\frac{1}{4}n^{2}\left(n+1\right)^{2}}$
$\displaystyle{\sum_{k=1}^{n}k^{4}=\frac{1}{30}\left(6n^{5}+15n^{4}+10n^{3}-n\right)=\frac{1}{30}n\left(n+1\right)\left(2n+1\right)\left(3n^{2}+3n-1\right)}$
$\displaystyle{\sum_{k=1}^{n}k^{5}=\frac{1}{12}\left(2n^{6}+6n^{5}+5n^{4}-n^{2}\right)=\frac{1}{12}n^{2}\left(n+1\right)^{2}\left(2n^{2}+2n-1\right)}$
$\displaystyle{\sum_{k=1}^{n}k^{6}=\frac{1}{42}\left(6n^{7}+2n^{6}+2n^{5}-7n^{3}+n\right)=\frac{1}{42}n\left(n+1\right)\left(2n+1\right)\left(3n^{4}+6n^{3}-3n+1\right)}$
$\displaystyle{\sum_{k=1}^{n}k^{7}=\frac{1}{24}\left(3n^{8}+12n^{7}+14n^{6}-7n^{4}+2n^{2}\right)=\frac{1}{24}n^{2}\left(n+1\right)^{2}\left(3n^{4}+6n^{3}-n^{2}-4n+2\right)}$
$\displaystyle{\sum_{k=1}^{n}k^{8}=\frac{1}{90}\left(10n^{9}+45n^{8}+60n^{7}-42n^{5}+20n^{3}-3n\right)=\frac{1}{90}n\left(n+1\right)\left(2n+1\right)\left(5n^{6}+15n^{5}+5n^{4}-15n^{3}-n^{2}+9n-3\right)}$
$\displaystyle{\sum_{k=1}^{n}k^{9}=\frac{1}{20}\left(2n^{10}+10n^{9}+15n^{8}-14n^{6}+10n^{4}-3n^{2}\right)=\frac{1}{20}n^{2}\left(n+1\right)^{2}\left(n^{2}+n-1\right)\left(2n^{4}+4n^{3}-n^{2}-3n+3\right)}$
$\displaystyle{\sum_{k=1}^{n}k^{10}=\frac{1}{66}\left(6n^{11}+33n^{10}+55n^{9}-66n^{7}+66n^{5}-33n^{3}+5n\right)=\frac{1}{66}n\left(n+1\right)\left(2n+1\right)\left(n^{2}+n-1\right)\left(3n^{6}+9n^{5}+2n^{4}-11n^{3}+3n^{2}+10n-5\right)}$
$\displaystyle{\sum_{k=1}^{n}k^{11}=\frac{1}{24}\left(2n^{12}+12n^{11}+22n^{10}-33n^{8}+44n^{6}-33n^{4}+10n^{2}\right)=\frac{1}{24}n^{2}\left(n+1\right)^{2}\left(2n^{8}+8n^{7}+4n^{6}-16n^{5}-5n^{4}+26n^{3}-3n^{2}-20n+10\right)}$
$\displaystyle{\sum_{k=1}^{n}k^{12}=\frac{1}{2730}\left(210n^{13}+1365n^{12}+2730n^{11}-5005n^{9}+8580n^{7}-9009n^{5}+4550n^{3}-69n\right)=\frac{1}{2730}n\left(n+1\right)\left(2n+1\right)\left(105n^{10}+525n^{9}+525n^{8}-1050n^{7}-1190n^{6}+2310n^{5}+1420n^{4}-3285n^{3}-287n^{2}+2073n-691\right)}$
$\displaystyle{\sum_{k=1}^{n}k^{13}=\frac{1}{420}\left(30n^{14}+210n^{13}+455n^{12}-100n^{10}+2145n^{8}-3003n^{6}+2275n^{4}-69n^{2}\right)=\frac{1}{420}n^{2}\left(n+1\right)^{2}\left(30n^{10}+150n^{9}+125n^{8}-400n^{7}-326n^{6}+1052n^{5}+367n^{4}-1786n^{3}+202n^{2}+1382n-691\right)}$
$\displaystyle{\sum_{k=1}^{n}k^{14}=\frac{1}{90}\left(6n^{15}+45n^{14}+105n^{13}-273n^{11}+715n^{9}-1287n^{7}+1365n^{5}-69n^{3}+105n\right)=\frac{1}{90}n\left(n+1\right)\left(2n+1\right)\left(3n^{12}+18n^{11}+24n^{10}-45n^{9}-81n^{8}+144n^{7}+182n^{6}-345n^{5}-217n^{4}+498n^{3}+44n^{2}-315n+105\right)}$
$\displaystyle{\sum_{k=1}^{n}k^{15}=\frac{1}{48}\left(3n^{16}+24n^{15}+60n^{14}-182n^{12}+572n^{10}-1287n^{8}+1820n^{6}-1382n^{4}+420n^{2}\right)=\frac{1}{48}n^{2}\left(n+1\right)^{2}\left(3n^{12}+18n^{11}+21n^{10}-60n^{9}-83n^{8}+226n^{7}+203n^{6}-632n^{5}-226n^{4}+1084n^{3}-122n^{2}-840n+420\right)}$
$\displaystyle{\sum_{k=1}^{n}k^{16}=\frac{1}{510}\left(30n^{17}+255n^{16}+680n^{15}-2380n^{13}+8840n^{11}-24310n^{9}+44200n^{7}-46988n^{5}+23800n^{3}-3617n\right)=\frac{1}{510}n\left(n+1\right)\left(2n+1\right)\left(15n^{14}+105n^{13}+175n^{12}-315n^{11}-805n^{10}+1365n^{9}+2775n^{8}-4845n^{7}-6275n^{6}+11835n^{5}+7485n^{4}-17145n^{3}-1519n^{2}+10851n-3617\right)}$
$\displaystyle{\sum_{k=1}^{n}k^{17}=\frac{1}{180}\left(10n^{18}+90n^{17}+255n^{16}-1020n^{14}+4420n^{12}-14586n^{10}+33150n^{8}-46988n^{6}+35700n^{4}-1085n^{2}\right)=\frac{1}{180}n^{2}\left(n+1\right)^{2}\left(10n^{14}+70n^{13}+105n^{12}-280n^{11}-565n^{10}+1410n^{9}+2165n^{8}-5740n^{7}-5271n^{6}+16282n^{5}+5857n^{4}-27996n^{3}+3147n^{2}+21702n-10851\right)}$
$\displaystyle{\sum_{k=1}^{n}k^{18}=\frac{1}{3990}\left(210n^{19}+1995n^{18}+5985n^{17}-27132n^{15}+135660n^{13}-529074n^{11}+1469650n^{9}-2678316n^{7}+2848860n^{5}-1443183n^{3}+219335n\right)=\frac{1}{3990}n\left(n+1\right)\left(2n+1\right)\left(105n^{16}+840n^{15}+1680n^{14}-2940n^{13}-9996n^{12}+16464n^{11}+48132n^{10}-80430n^{9}-167958n^{8}+292152n^{7}+380576n^{6}-716940n^{5}-454036n^{4}+1039524n^{3}+92162n^{2}-658005n+219335\right)}$
$\displaystyle{\sum_{k=1}^{n}k^{19}=\frac{1}{840}\left(42n^{20}+420n^{19}+1330n^{18}-6783n^{16}+38760n^{14}-176358n^{12}+587860n^{10}-1339158n^{8}+1899240n^{6}-1443183n^{4}+438670n^{2}\right)=\frac{1}{840}n^{2}\left(n+1\right)^{2}\left(42n^{16}+336n^{15}+616n^{14}-1568n^{13}-4263n^{12}+10094n^{11}+22835n^{10}-55764n^{9}-87665n^{8}+231094n^{7}+213337n^{6}-657768n^{5}-236959n^{4}+1131686n^{3}-127173n^{2}-877340n+438670\right)}$
$\displaystyle{\sum_{k=1}^{n}k^{20}=\frac{1}{6930}\left(330n^{21}+3465n^{20}+11550n^{19}-65835n^{17}+426360n^{15}-2238390n^{13}+8817900n^{11}-24551230n^{9}+44767800n^{7}-47625039n^{5}+24126850n^{3}-366683n\right)=\frac{1}{6930}n\left(n+1\right)\left(2n+1\right)\left(165n^{18}+1485n^{17}+3465n^{16}-5940n^{15}-25740n^{14}+41580n^{13}+163680n^{12}-266310n^{11}-801570n^{10}+1335510n^{9}+2806470n^{8}-4877460n^{7}-6362660n^{6}+11982720n^{5}+7591150n^{4}-17378085n^{3}-1540967n^{2}+11000493n-3666831\right)}$