Explicit Expressions, and Limiting Scaled Moments, for the first, 8, moments of Sir Tony Hoare's Famous Quicksort Algorithm By Shalosh B. Ekhad Quicksort is one of the most commonly used sorting algorithms invented by To\ ny Hoare in 1959 when he was about 25 years old. The expected number of comparisons is very well-known, but the variance, is \ less well-known and as far as we know, there are no explicit expressions for the higher moments. Here we will state\ all of them up to the, 8, -th one. But before, we need to define some notation Definition n ----- \ 1 H[m](n) = ) ---- / m ----- i i = 1 The following theorem can be found in many textbooks, and wikipedia, but we \ discovered it from scratch. Theorem 1: The expected number of comparisons in Quicksort is -4 n + H[1](n) (2 n + 2) and in Maple format -4*n+H[1](n)*(2*n+2) and in LaTex -4\,n+H_{{1}} \left( n \right) \left( 2\,n+2 \right) This is asymptotically (-4 + 2 ln(n) + 2 gamma) n + 2 ln(n) + 2 gamma and in Maple format (-4+2*ln(n)+2*gamma)*n+2*ln(n)+2*gamma and in LaTex \left( -4+2\,\ln \left( n \right) +2\,\gamma \right) n+2\,\ln \left( n \right) +2\,\gamma In floating point this is (-2.8455686701969342788 + 2. ln(n)) n + 2. ln(n) + 1.1544313298030657212 written in Maple format this is (-2.8455686701969342788+2.*ln(n))*n+2.*ln(n)+1.1544313298030657212 and in LaTex (-2.8455686701969342788+2.*ln(n))*n+2.*ln(n)+1.1544313298030657212 The following theorem, about the variance is less well known. It is found in "Quicksort algorithm again revisited" by Charles Knessl and Wojciech Szpank\ owski that appeared in Discrete Matehmatics and Theoretical Computer Science, v. 3 (1999), 43-64, E\ q. (32) Theorem 2: The variance of the random variable "number of comparisons in Qui\ cksort applied to lists of length n" is 2 n (7 n + 13) + H[1](n) (-2 n - 2) - 4 (n + 1) H[2](n) and in Maple format n*(7*n+13)+H[1](n)*(-2*n-2)-4*(n+1)^2*H[2](n) and in LaTex n \left( 7\,n+13 \right) +H_{{1}} \left( n \right) \left( -2\,n-2 \right) -4\, \left( n+1 \right) ^{2}H_{{2}} \left( n \right) This is asymptotically / 2\ / 2\ 2 | 2 Pi | 2 | 4 Pi | 2 Pi |7 - -----| n + |13 - 2 ln(n) - 2 gamma - -----| n - 2 ln(n) - 2 gamma - ----- \ 3 / \ 3 / 3 For the leading term, this agrees with Eq. (2.8) (p. 198) of "A note concern\ ing the limiting distribution of the quicksort algorithm" by Michael Cramer, Informatique Theorique et Applications, v. 20 (1996), 195\ -207. and in Maple format (7-2/3*Pi^2)*n^2+(13-2*ln(n)-2*gamma-4/3*Pi^2)*n-2*ln(n)-2*gamma-2/3*Pi^2 and in LaTex \left( 7-2/3\,{\pi }^{2} \right) {n}^{2}+ \left( 13-2\,\ln \left( n \right) -2\,\gamma-4/3\,{\pi }^{2} \right) n-2\,\ln \left( n \right) -2\,\gamma-2/3\,{\pi }^{2} In floating point this is 2 0.4202637326070942539 n + (-1.313903864588877213 - 2. ln(n)) n - 2. ln(n) - 7.7341675971959714673 written in Maple format this is .4202637326070942539*n^2+(-1.313903864588877213-2.*ln(n))*n-2.*ln(n)-7.73416759\ 71959714673 and in LaTex .4202637326070942539*n^2+(-1.313903864588877213-2.*ln(n))*n-2.*ln(n)-7.73416759\ 71959714673 The remaining, 6, theorems seem to be new, except for the leading terms for t\ he third moment given in Eq. (2.9) of the above-mentioned paper by M. Cramer. Cramer also estimated, numerically, \ the leading term for the fourth moment, but here we have an explicit expression in terms of Pi. Theorem No. , 3, : The , 3, -th moment about the mean of the random variabl\ e "number of comparisons in Quicksort applied to lists of length n" is 2 2 -n (19 n + 81 n + 104) + H[1](n) (14 n + 14) + 12 (n + 1) H[2](n) 3 + 16 (n + 1) H[3](n) and in Maple format -n*(19*n^2+81*n+104)+H[1](n)*(14*n+14)+12*(n+1)^2*H[2](n)+16*(n+1)^3*H[3](n) and in LaTex -n \left( 19\,{n}^{2}+81\,n+104 \right) +H_{{1}} \left( n \right) \left( 14\,n+14 \right) +12\, \left( n+1 \right) ^{2}H_{{2}} \left( n \right) +16\, \left( n+1 \right) ^{3}H_{{3}} \left( n \right) This is asymptotically 3 2 2 (-19 + 16 Zeta(3)) n + (-81 + 2 Pi + 48 Zeta(3)) n 2 + (-104 + 14 ln(n) + 14 gamma + 4 Pi + 48 Zeta(3)) n + 14 ln(n) 2 + 14 gamma + 2 Pi + 16 Zeta(3) and in Maple format (-19+16*Zeta(3))*n^3+(-81+2*Pi^2+48*Zeta(3))*n^2+(-104+14*ln(n)+14*gamma+4*Pi^2 +48*Zeta(3))*n+14*ln(n)+14*gamma+2*Pi^2+16*Zeta(3) and in LaTex \left( -19+16\,\zeta \left( 3 \right) \right) {n}^{3}+ \left( -81+2 \,{\pi }^{2}+48\,\zeta \left( 3 \right) \right) {n}^{2}+ \left( -104+ 14\,\ln \left( n \right) +14\,\gamma+4\,{\pi }^{2}+48\,\zeta \left( 3 \right) \right) n+14\,\ln \left( n \right) +14\,\gamma+2\,{\pi }^{2} +16\,\zeta \left( 3 \right) In floating point this is 3 2 0.232910450553508566 n - 3.562059846160757063 n + (1.258168264639420223 + 14. ln(n)) n + 14. ln(n) + 47.053138561353685852 written in Maple format this is .232910450553508566*n^3-3.562059846160757063*n^2+(1.258168264639420223+14.*ln(n ))*n+14.*ln(n)+47.053138561353685852 and in LaTex .232910450553508566*n^3-3.562059846160757063*n^2+(1.258168264639420223+14.*ln(n ))*n+14.*ln(n)+47.053138561353685852 The limit of the scaled , 3, -th moment is -19 + 16 Zeta(3) ---------------- / 2\3/2 | 2 Pi | |7 - -----| \ 3 / in Maple format this is: (-19+16*Zeta(3))/(7-2/3*Pi^2)^(3/2) in LaTex this is: {\frac {-19+16\,\zeta \left( 3 \right) }{ \left( 7-2/3\,{\pi }^{2} \right) ^{3/2}}} In floating point, this is , 3, -th moment is 0.85488186713258853660 in Maple format this is: .85488186713258853660 in LaTex this is: 0.85488186713258853660 Theorem No. , 4, : The , 4, -th moment about the mean of the random variabl\ e "number of comparisons in Quicksort applied to lists of length n" is 3 2 n (2260 n + 9658 n + 15497 n + 11357) 2 --------------------------------------- - 2 (n + 1) (42 n + 78 n + 77) H[1](n) 9 2 2 + 12 (n + 1) H[1](n) 2 2 3 + (-4 (42 n + 78 n + 31) (n + 1) + 48 (n + 1) H[1](n)) H[2](n) 4 2 3 4 + 48 (n + 1) H[2](n) - 96 (n + 1) H[3](n) - 96 (n + 1) H[4](n) and in Maple format 1/9*n*(2260*n^3+9658*n^2+15497*n+11357)-2*(n+1)*(42*n^2+78*n+77)*H[1](n)+12*(n+ 1)^2*H[1](n)^2+(-4*(42*n^2+78*n+31)*(n+1)^2+48*(n+1)^3*H[1](n))*H[2](n)+48*(n+1 )^4*H[2](n)^2-96*(n+1)^3*H[3](n)-96*(n+1)^4*H[4](n) and in LaTex 1/9\,n \left( 2260\,{n}^{3}+9658\,{n}^{2}+15497\,n+11357 \right) -2\, \left( n+1 \right) \left( 42\,{n}^{2}+78\,n+77 \right) H_{{1}} \left( n \right) +12\, \left( n+1 \right) ^{2} \left( H_{{1}} \left( n \right) \right) ^{2}+ \left( -4\, \left( 42\,{n}^{2}+78\,n+31 \right) \left( n+1 \right) ^{2}+48\, \left( n+1 \right) ^{3}H_{{1}} \left( n \right) \right) H_{{2}} \left( n \right) +48\, \left( n+1 \right) ^{4} \left( H_{{2}} \left( n \right) \right) ^{2}-96\, \left( n+1 \right) ^{3}H_{{3}} \left( n \right) -96\, \left( n+1 \right) ^{4}H_{{4}} \left( n \right) This is asymptotically / 2 4 4 | (2260/9 - 28 Pi + 4/15 Pi ) n + |9658/9 - 84 ln(n) - 84 gamma \ 4 \ / 2 16 Pi | 3 | + 1/6 (-648 + 48 ln(n) + 48 gamma) Pi + ------ - 96 Zeta(3)| n + | 15 / \ 2 15497/9 - 240 ln(n) - 240 gamma + 12 (ln(n) + gamma) 4 \ / 2 8 Pi | 2 | + 1/6 (-916 + 144 ln(n) + 144 gamma) Pi + ----- - 288 Zeta(3)| n + | 5 / \ 2 11357/9 - 310 ln(n) - 310 gamma + 24 (ln(n) + gamma) 4 \ 2 16 Pi | + 1/6 (-560 + 144 ln(n) + 144 gamma) Pi + ------ - 288 Zeta(3)| n 15 / 2 - 154 ln(n) - 154 gamma + 12 (ln(n) + gamma) 4 2 4 Pi + 1/6 (-124 + 48 ln(n) + 48 gamma) Pi + ----- - 96 Zeta(3) 15 and in Maple format (2260/9-28*Pi^2+4/15*Pi^4)*n^4+(9658/9-84*ln(n)-84*gamma+1/6*(-648+48*ln(n)+48* gamma)*Pi^2+16/15*Pi^4-96*Zeta(3))*n^3+(15497/9-240*ln(n)-240*gamma+12*(ln(n)+ gamma)^2+1/6*(-916+144*ln(n)+144*gamma)*Pi^2+8/5*Pi^4-288*Zeta(3))*n^2+(11357/9 -310*ln(n)-310*gamma+24*(ln(n)+gamma)^2+1/6*(-560+144*ln(n)+144*gamma)*Pi^2+16/ 15*Pi^4-288*Zeta(3))*n-154*ln(n)-154*gamma+12*(ln(n)+gamma)^2+1/6*(-124+48*ln(n )+48*gamma)*Pi^2+4/15*Pi^4-96*Zeta(3) and in LaTex \left( {\frac {2260}{9}}-28\,{\pi }^{2}+{\frac {4}{15}}\,{\pi }^{4} \right) {n}^{4}+ \left( {\frac {9658}{9}}-84\,\ln \left( n \right) - 84\,\gamma+1/6\, \left( -648+48\,\ln \left( n \right) +48\,\gamma \right) {\pi }^{2}+{\frac {16}{15}}\,{\pi }^{4}-96\,\zeta \left( 3 \right) \right) {n}^{3}+ \left( {\frac {15497}{9}}-240\,\ln \left( n \right) -240\,\gamma+12\, \left( \ln \left( n \right) +\gamma \right) ^{2}+1/6\, \left( -916+144\,\ln \left( n \right) +144\,\gamma \right) {\pi }^{2}+8/5\,{\pi }^{4}-288\,\zeta \left( 3 \right) \right) {n}^{2}+ \left( {\frac {11357}{9}}-310\,\ln \left( n \right) -310\,\gamma+24\, \left( \ln \left( n \right) +\gamma \right) ^{2}+1/6 \, \left( -560+144\,\ln \left( n \right) +144\,\gamma \right) {\pi }^{ 2}+{\frac {16}{15}}\,{\pi }^{4}-288\,\zeta \left( 3 \right) \right) n -154\,\ln \left( n \right) -154\,\gamma+12\, \left( \ln \left( n \right) +\gamma \right) ^{2}+1/6\, \left( -124+48\,\ln \left( n \right) +48\,\gamma \right) {\pi }^{2}+{\frac {4}{15}}\,{\pi }^{4}-96 \,\zeta \left( 3 \right) In floating point this is 4 0.737945489676386378 n 3 + (-7.21159019180105191 - 5.043164791285131045 ln(n)) n + ( 22.98504802454367149 - 3.12949437385539313 ln(n) 2 2 + 12. (ln(n) + 0.57721566490153286061) ) n + (56.22496406127034937 2 - 73.12949437385539313 ln(n) + 24. (ln(n) + 0.57721566490153286061) ) n - 75.043164791285131045 ln(n) - 336.70961964475076028 2 + 12. (ln(n) + 0.57721566490153286061) written in Maple format this is .737945489676386378*n^4+(-7.21159019180105191-5.043164791285131045*ln(n))*n^3+( 22.98504802454367149-3.12949437385539313*ln(n)+12.*(ln(n)+.57721566490153286061 )^2)*n^2+(56.22496406127034937-73.12949437385539313*ln(n)+24.*(ln(n)+.577215664\ 90153286061)^2)*n-75.043164791285131045*ln(n)-336.70961964475076028+12.*(ln(n)+ .57721566490153286061)^2 and in LaTex .737945489676386378*n^4+(-7.21159019180105191-5.043164791285131045*ln(n))*n^3+( 22.98504802454367149-3.12949437385539313*ln(n)+12.*(ln(n)+.57721566490153286061 )^2)*n^2+(56.22496406127034937-73.12949437385539313*ln(n)+24.*(ln(n)+.577215664\ 90153286061)^2)*n-75.043164791285131045*ln(n)-336.70961964475076028+12.*(ln(n)+ .57721566490153286061)^2 The limit of the scaled , 4, -th moment is 2 4 2260/9 - 28 Pi + 4/15 Pi -------------------------- / 2\2 | 2 Pi | |7 - -----| \ 3 / in Maple format this is: (2260/9-28*Pi^2+4/15*Pi^4)/(7-2/3*Pi^2)^2 in LaTex this is: {\frac {{\frac {2260}{9}}-28\,{\pi }^{2}+{\frac {4}{15}}\,{\pi }^{4}}{ \left( 7-2/3\,{\pi }^{2} \right) ^{2}}} In floating point, this is , 4, -th moment is 4.1781156382698542397 in Maple format this is: 4.1781156382698542397 in LaTex this is: 4.1781156382698542397 Theorem No. , 5, : The , 5, -th moment about the mean of the random variabl\ e "number of comparisons in Quicksort applied to lists of length n" is 4 3 2 n (229621 n + 1422035 n + 3401325 n + 3915865 n + 2217794) - ------------------------------------------------------------- 108 3 2 + 2 (n + 1) (190 n + 1300 n + 1950 n + 1171) H[1](n) 2 2 - 280 (n + 1) H[1](n) + 3 2 2 3 (20 (38 n + 204 n + 286 n + 91) (n + 1) - 800 (n + 1) H[1](n)) H[2](n) 4 2 2 3 - 480 (n + 1) H[2](n) + (80 (14 n + 26 n + 17) (n + 1) 4 5 - 320 (n + 1) H[1](n) - 640 (n + 1) H[2](n)) H[3](n) 4 5 + 960 (n + 1) H[4](n) + 768 (n + 1) H[5](n) and in Maple format -1/108*n*(229621*n^4+1422035*n^3+3401325*n^2+3915865*n+2217794)+2*(n+1)*(190*n^ 3+1300*n^2+1950*n+1171)*H[1](n)-280*(n+1)^2*H[1](n)^2+(20*(38*n^3+204*n^2+286*n +91)*(n+1)^2-800*(n+1)^3*H[1](n))*H[2](n)-480*(n+1)^4*H[2](n)^2+(80*(14*n^2+26* n+17)*(n+1)^3-320*(n+1)^4*H[1](n)-640*(n+1)^5*H[2](n))*H[3](n)+960*(n+1)^4*H[4] (n)+768*(n+1)^5*H[5](n) and in LaTex -{\frac {1}{108}}\,n \left( 229621\,{n}^{4}+1422035\,{n}^{3}+3401325\,{ n}^{2}+3915865\,n+2217794 \right) +2\, \left( n+1 \right) \left( 190\, {n}^{3}+1300\,{n}^{2}+1950\,n+1171 \right) H_{{1}} \left( n \right) - 280\, \left( n+1 \right) ^{2} \left( H_{{1}} \left( n \right) \right) ^{2}+ \left( 20\, \left( 38\,{n}^{3}+204\,{n}^{2}+286\,n+91 \right) \left( n+1 \right) ^{2}-800\, \left( n+1 \right) ^{3}H_{{1}} \left( n \right) \right) H_{{2}} \left( n \right) -480\, \left( n+1 \right) ^{ 4} \left( H_{{2}} \left( n \right) \right) ^{2}+ \left( 80\, \left( 14 \,{n}^{2}+26\,n+17 \right) \left( n+1 \right) ^{3}-320\, \left( n+1 \right) ^{4}H_{{1}} \left( n \right) -640\, \left( n+1 \right) ^{5}H_{ {2}} \left( n \right) \right) H_{{3}} \left( n \right) +960\, \left( n +1 \right) ^{4}H_{{4}} \left( n \right) +768\, \left( n+1 \right) ^{5}H _{{5}} \left( n \right) This is asymptotically / 2 / 2\ \ / | 229621 380 Pi | 320 Pi | | 5 | 1422035 |- ------ + ------- + |1120 - -------| Zeta(3) + 768 Zeta(5)| n + |- ------- \ 108 3 \ 3 / / \ 108 2 4 2800 Pi 8 Pi + 380 ln(n) + 380 gamma + -------- - ----- 3 3 / 2\ \ / | 1600 Pi | | 4 | + |5440 - 320 ln(n) - 320 gamma - --------| Zeta(3) + 3840 Zeta(5)| n + | \ 3 / / \ - 125975/4 + 2980 ln(n) + 2980 gamma 4 2 32 Pi + 1/6 (14640 - 800 ln(n) - 800 gamma) Pi - ------ 3 / 2\ \ | 3200 Pi | | 3 + |10960 - 1280 ln(n) - 1280 gamma - --------| Zeta(3) + 7680 Zeta(5)| n \ 3 / / / | 3915865 2 + |- ------- + 6500 ln(n) + 6500 gamma - 280 (ln(n) + gamma) \ 108 2 4 + 1/6 (17340 - 2400 ln(n) - 2400 gamma) Pi - 16 Pi / 2\ \ | 3200 Pi | | 2 + |11440 - 1920 ln(n) - 1920 gamma - --------| Zeta(3) + 7680 Zeta(5)| n \ 3 / / / | 1108897 2 + |- ------- + 6242 ln(n) + 6242 gamma - 560 (ln(n) + gamma) \ 54 4 2 32 Pi + 1/6 (9360 - 2400 ln(n) - 2400 gamma) Pi - ------ 3 / 2\ \ | 1600 Pi | | + |6160 - 1280 ln(n) - 1280 gamma - --------| Zeta(3) + 3840 Zeta(5)| n \ 3 / / 2 + 2342 ln(n) + 2342 gamma - 280 (ln(n) + gamma) 4 2 8 Pi + 1/6 (1820 - 800 ln(n) - 800 gamma) Pi - ----- 3 / 2\ | 320 Pi | + |1360 - 320 ln(n) - 320 gamma - -------| Zeta(3) + 768 Zeta(5) \ 3 / and in Maple format (-229621/108+380/3*Pi^2+(1120-320/3*Pi^2)*Zeta(3)+768*Zeta(5))*n^5+(-1422035/ 108+380*ln(n)+380*gamma+2800/3*Pi^2-8/3*Pi^4+(5440-320*ln(n)-320*gamma-1600/3* Pi^2)*Zeta(3)+3840*Zeta(5))*n^4+(-125975/4+2980*ln(n)+2980*gamma+1/6*(14640-800 *ln(n)-800*gamma)*Pi^2-32/3*Pi^4+(10960-1280*ln(n)-1280*gamma-3200/3*Pi^2)*Zeta (3)+7680*Zeta(5))*n^3+(-3915865/108+6500*ln(n)+6500*gamma-280*(ln(n)+gamma)^2+1 /6*(17340-2400*ln(n)-2400*gamma)*Pi^2-16*Pi^4+(11440-1920*ln(n)-1920*gamma-3200 /3*Pi^2)*Zeta(3)+7680*Zeta(5))*n^2+(-1108897/54+6242*ln(n)+6242*gamma-560*(ln(n )+gamma)^2+1/6*(9360-2400*ln(n)-2400*gamma)*Pi^2-32/3*Pi^4+(6160-1280*ln(n)-\ 1280*gamma-1600/3*Pi^2)*Zeta(3)+3840*Zeta(5))*n+2342*ln(n)+2342*gamma-280*(ln(n )+gamma)^2+1/6*(1820-800*ln(n)-800*gamma)*Pi^2-8/3*Pi^4+(1360-320*ln(n)-320* gamma-320/3*Pi^2)*Zeta(3)+768*Zeta(5) and in LaTex \left( -{\frac {229621}{108}}+{\frac {380}{3}}\,{\pi }^{2}+ \left( 1120-{\frac {320}{3}}\,{\pi }^{2} \right) \zeta \left( 3 \right) +768 \,\zeta \left( 5 \right) \right) {n}^{5}+ \left( -{\frac {1422035}{ 108}}+380\,\ln \left( n \right) +380\,\gamma+{\frac {2800}{3}}\,{\pi } ^{2}-8/3\,{\pi }^{4}+ \left( 5440-320\,\ln \left( n \right) -320\, \gamma-{\frac {1600}{3}}\,{\pi }^{2} \right) \zeta \left( 3 \right) + 3840\,\zeta \left( 5 \right) \right) {n}^{4}+ \left( -{\frac {125975} {4}}+2980\,\ln \left( n \right) +2980\,\gamma+1/6\, \left( 14640-800\, \ln \left( n \right) -800\,\gamma \right) {\pi }^{2}-{\frac {32}{3}}\, {\pi }^{4}+ \left( 10960-1280\,\ln \left( n \right) -1280\,\gamma-{ \frac {3200}{3}}\,{\pi }^{2} \right) \zeta \left( 3 \right) +7680\, \zeta \left( 5 \right) \right) {n}^{3}+ \left( -{\frac {3915865}{108} }+6500\,\ln \left( n \right) +6500\,\gamma-280\, \left( \ln \left( n \right) +\gamma \right) ^{2}+1/6\, \left( 17340-2400\,\ln \left( n \right) -2400\,\gamma \right) {\pi }^{2}-16\,{\pi }^{4}+ \left( 11440- 1920\,\ln \left( n \right) -1920\,\gamma-{\frac {3200}{3}}\,{\pi }^{2} \right) \zeta \left( 3 \right) +7680\,\zeta \left( 5 \right) \right) {n}^{2}+ \left( -{\frac {1108897}{54}}+6242\,\ln \left( n \right) +6242\,\gamma-560\, \left( \ln \left( n \right) +\gamma \right) ^{2}+1/6\, \left( 9360-2400\,\ln \left( n \right) -2400\, \gamma \right) {\pi }^{2}-{\frac {32}{3}}\,{\pi }^{4}+ \left( 6160-1280 \,\ln \left( n \right) -1280\,\gamma-{\frac {1600}{3}}\,{\pi }^{2} \right) \zeta \left( 3 \right) +3840\,\zeta \left( 5 \right) \right) n+2342\,\ln \left( n \right) +2342\,\gamma-280\, \left( \ln \left( n \right) +\gamma \right) ^{2}+1/6\, \left( 1820-800\,\ln \left( n \right) -800\,\gamma \right) {\pi }^{2}-8/3\,{\pi }^{4}+ \left( 1360-320\,\ln \left( n \right) -320\,\gamma-{\frac {320}{3}}\, {\pi }^{2} \right) \zeta \left( 3 \right) +768\,\zeta \left( 5 \right) In floating point this is 5 4 1.21898373286590873 n + (-24.1881217055351921 - 4.65820901107017133 ln(n)) n 3 + (104.8497475408336461 + 125.4199104771381655 ln(n)) n + ( -92.0484549718454146 + 244.2089854978355243 ln(n) 2 2 - 280. (ln(n) + 0.57721566490153286061) ) n + (-682.3767878878473255 2 + 755.5254035199758670 ln(n) - 560. (ln(n) + 0.57721566490153286061) ) n + 641.3945375103486795 ln(n) + 4269.9285200632328364 2 - 280. (ln(n) + 0.57721566490153286061) written in Maple format this is 1.21898373286590873*n^5+(-24.1881217055351921-4.65820901107017133*ln(n))*n^4+( 104.8497475408336461+125.4199104771381655*ln(n))*n^3+(-92.0484549718454146+244.\ 2089854978355243*ln(n)-280.*(ln(n)+.57721566490153286061)^2)*n^2+(-682.37678788\ 78473255+755.5254035199758670*ln(n)-560.*(ln(n)+.57721566490153286061)^2)*n+641\ .3945375103486795*ln(n)+4269.9285200632328364-280.*(ln(n)+.57721566490153286061 )^2 and in LaTex 1.21898373286590873*n^5+(-24.1881217055351921-4.65820901107017133*ln(n))*n^4+( 104.8497475408336461+125.4199104771381655*ln(n))*n^3+(-92.0484549718454146+244.\ 2089854978355243*ln(n)-280.*(ln(n)+.57721566490153286061)^2)*n^2+(-682.37678788\ 78473255+755.5254035199758670*ln(n)-560.*(ln(n)+.57721566490153286061)^2)*n+641\ .3945375103486795*ln(n)+4269.9285200632328364-280.*(ln(n)+.57721566490153286061 )^2 The limit of the scaled , 5, -th moment is 2 / 2\ 229621 380 Pi | 320 Pi | - ------ + ------- + |1120 - -------| Zeta(3) + 768 Zeta(5) 108 3 \ 3 / ----------------------------------------------------------- / 2\5/2 | 2 Pi | |7 - -----| \ 3 / in Maple format this is: (-229621/108+380/3*Pi^2+(1120-320/3*Pi^2)*Zeta(3)+768*Zeta(5))/(7-2/3*Pi^2)^(5/ 2) in LaTex this is: {\frac {-{\frac {229621}{108}}+{\frac {380}{3}}\,{\pi }^{2}+ \left( 1120-{\frac {320}{3}}\,{\pi }^{2} \right) \zeta \left( 3 \right) +768 \,\zeta \left( 5 \right) }{ \left( 7-2/3\,{\pi }^{2} \right) ^{5/2}}} In floating point, this is , 5, -th moment is 10.646163374673878503 in Maple format this is: 10.646163374673878503 in LaTex this is: 10.646163374673878503 Theorem No. , 6, : The , 6, -th moment about the mean of the random variabl\ e "number of comparisons in Quicksort applied to lists of length n" is 5 4 3 2 n (74250517 n + 523547007 n + 1579578725 n + 2571768745 n + 2342670258 n + 1133389148)/2700 4 3 2 - 2/3 (n + 1) (11300 n + 56270 n + 135760 n + 145510 n + 68427) H[1](n) / 2 2 2 3 3 | + 20 (63 n + 117 n + 329) (n + 1) H[1](n) - 120 (n + 1) H[1](n) + | \ 4 3 2 2 4 (11300 n + 51710 n + 101830 n + 93640 n + 26013) (n + 1) - -------------------------------------------------------------- 3 \ 2 3 4 2| + 240 (21 n + 39 n + 68) (n + 1) H[1](n) - 720 (n + 1) H[1](n) | / H[2](n) 2 4 5 2 + (240 (21 n + 39 n + 37) (n + 1) - 1440 (n + 1) H[1](n)) H[2](n) 6 3 3 2 3 - 960 (n + 1) H[2](n) + (-160 (38 n + 225 n + 325 n + 159) (n + 1) 4 5 + 7360 (n + 1) H[1](n) + 9600 (n + 1) H[2](n)) H[3](n) 6 2 2 4 + 2560 (n + 1) H[3](n) + (-480 (21 n + 39 n + 37) (n + 1) 5 6 + 2880 (n + 1) H[1](n) + 5760 (n + 1) H[2](n)) H[4](n) 5 6 - 11520 (n + 1) H[5](n) - 7680 (n + 1) H[6](n) and in Maple format 1/2700*n*(74250517*n^5+523547007*n^4+1579578725*n^3+2571768745*n^2+2342670258*n +1133389148)-2/3*(n+1)*(11300*n^4+56270*n^3+135760*n^2+145510*n+68427)*H[1](n)+ 20*(63*n^2+117*n+329)*(n+1)^2*H[1](n)^2-120*(n+1)^3*H[1](n)^3+(-4/3*(11300*n^4+ 51710*n^3+101830*n^2+93640*n+26013)*(n+1)^2+240*(21*n^2+39*n+68)*(n+1)^3*H[1](n )-720*(n+1)^4*H[1](n)^2)*H[2](n)+(240*(21*n^2+39*n+37)*(n+1)^4-1440*(n+1)^5*H[1 ](n))*H[2](n)^2-960*(n+1)^6*H[2](n)^3+(-160*(38*n^3+225*n^2+325*n+159)*(n+1)^3+ 7360*(n+1)^4*H[1](n)+9600*(n+1)^5*H[2](n))*H[3](n)+2560*(n+1)^6*H[3](n)^2+(-480 *(21*n^2+39*n+37)*(n+1)^4+2880*(n+1)^5*H[1](n)+5760*(n+1)^6*H[2](n))*H[4](n)-\ 11520*(n+1)^5*H[5](n)-7680*(n+1)^6*H[6](n) and in LaTex {\frac {1}{2700}}\,n \left( 74250517\,{n}^{5}+523547007\,{n}^{4}+ 1579578725\,{n}^{3}+2571768745\,{n}^{2}+2342670258\,n+1133389148 \right) -2/3\, \left( n+1 \right) \left( 11300\,{n}^{4}+56270\,{n}^{3 }+135760\,{n}^{2}+145510\,n+68427 \right) H_{{1}} \left( n \right) +20 \, \left( 63\,{n}^{2}+117\,n+329 \right) \left( n+1 \right) ^{2} \left( H_{{1}} \left( n \right) \right) ^{2}-120\, \left( n+1 \right) ^{3} \left( H_{{1}} \left( n \right) \right) ^{3}+ \left( -4/ 3\, \left( 11300\,{n}^{4}+51710\,{n}^{3}+101830\,{n}^{2}+93640\,n+26013 \right) \left( n+1 \right) ^{2}+240\, \left( 21\,{n}^{2}+39\,n+68 \right) \left( n+1 \right) ^{3}H_{{1}} \left( n \right) -720\, \left( n+1 \right) ^{4} \left( H_{{1}} \left( n \right) \right) ^{2} \right) H_{{2}} \left( n \right) + \left( 240\, \left( 21\,{n}^{2}+39 \,n+37 \right) \left( n+1 \right) ^{4}-1440\, \left( n+1 \right) ^{5}H _{{1}} \left( n \right) \right) \left( H_{{2}} \left( n \right) \right) ^{2}-960\, \left( n+1 \right) ^{6} \left( H_{{2}} \left( n \right) \right) ^{3}+ \left( -160\, \left( 38\,{n}^{3}+225\,{n}^{2}+ 325\,n+159 \right) \left( n+1 \right) ^{3}+7360\, \left( n+1 \right) ^ {4}H_{{1}} \left( n \right) +9600\, \left( n+1 \right) ^{5}H_{{2}} \left( n \right) \right) H_{{3}} \left( n \right) +2560\, \left( n+1 \right) ^{6} \left( H_{{3}} \left( n \right) \right) ^{2}+ \left( - 480\, \left( 21\,{n}^{2}+39\,n+37 \right) \left( n+1 \right) ^{4}+2880 \, \left( n+1 \right) ^{5}H_{{1}} \left( n \right) +5760\, \left( n+1 \right) ^{6}H_{{2}} \left( n \right) \right) H_{{4}} \left( n \right) -11520\, \left( n+1 \right) ^{5}H_{{5}} \left( n \right) -7680 \, \left( n+1 \right) ^{6}H_{{6}} \left( n \right) This is asymptotically / 2 6 |74250517 22600 Pi 4 88 Pi 2 |-------- - --------- + 140 Pi - ------ - 6080 Zeta(3) + 2560 Zeta(3) \ 2700 9 7 2 4\ / (960 Pi - 10080) Pi | 6 |174515669 22600 gamma + ---------------------| n + |--------- - 22600/3 ln(n) - ----------- 90 / \ 900 3 2 + 1/6 (-99080 + 5040 ln(n) + 5040 gamma) Pi 6 4 528 Pi + 1/36 (29520 - 1440 ln(n) - 1440 gamma) Pi - ------- 7 2 2 + (1600 Pi - 54240) Zeta(3) + 15360 Zeta(3) \ 2 4 | + 1/90 (5760 Pi + 2880 ln(n) + 2880 gamma - 59040) Pi - 11520 Zeta(5)| / / 5 |63183149 135140 gamma 2 n + |-------- - 135140/3 ln(n) - ------------ + 1260 (ln(n) + gamma) \ 108 3 2 2 + 1/6 (- 866200/3 + 24480 ln(n) + 24480 gamma - 720 (ln(n) + gamma) ) Pi 6 4 1320 Pi + 1/36 (76560 - 7200 ln(n) - 7200 gamma) Pi - -------- 7 2 2 + (8000 Pi + 7360 ln(n) + 7360 gamma - 178240) Zeta(3) + 38400 Zeta(3) 2 4 + 1/90 (14400 Pi + 14400 ln(n) + 14400 gamma - 153120) Pi \ / | 4 |514353749 - 57600 Zeta(5)| n + |--------- - 128020 ln(n) - 128020 gamma / \ 540 2 3 + 4860 (ln(n) + gamma) - 120 (ln(n) + gamma) + 2 2 1/6 (- 1396040/3 + 59520 ln(n) + 59520 gamma - 2880 (ln(n) + gamma) ) Pi 6 4 1760 Pi + 1/36 (111840 - 14400 ln(n) - 14400 gamma) Pi - -------- 7 2 + (16000 Pi + 29440 ln(n) + 29440 gamma - 295520) Zeta(3) 2 + 51200 Zeta(3) 2 4 + 1/90 (19200 Pi + 28800 ln(n) + 28800 gamma - 223680) Pi \ / | 3 |390445043 562540 gamma - 115200 Zeta(5)| n + |--------- - 562540/3 ln(n) - ------------ / \ 450 3 2 3 + 12520 (ln(n) + gamma) - 360 (ln(n) + gamma) 2 2 + 1/6 (-4320 (ln(n) + gamma) + 82080 ln(n) + 82080 gamma - 420164) Pi 6 4 1320 Pi + 1/36 (95760 - 14400 ln(n) - 14400 gamma) Pi - -------- 7 2 + (16000 Pi + 44160 ln(n) + 44160 gamma - 268320) Zeta(3) 2 + 38400 Zeta(3) 2 4 + 1/90 (14400 Pi + 28800 ln(n) + 28800 gamma - 191520) Pi \ / | 2 |283347287 427874 gamma - 115200 Zeta(5)| n + |--------- - 427874/3 ln(n) - ------------ / \ 675 3 2 3 + 15500 (ln(n) + gamma) - 360 (ln(n) + gamma) 2 2 + 1/6 (- 582664/3 + 58320 ln(n) + 58320 gamma - 2880 (ln(n) + gamma) ) Pi 6 4 528 Pi + 1/36 (44880 - 7200 ln(n) - 7200 gamma) Pi - ------- 7 2 2 + (8000 Pi + 29440 ln(n) + 29440 gamma - 128320) Zeta(3) + 15360 Zeta(3) \ 2 4 | + 1/90 (5760 Pi + 14400 ln(n) + 14400 gamma - 89760) Pi - 57600 Zeta(5)| / 2 n - 45618 ln(n) - 45618 gamma + 6580 (ln(n) + gamma) 3 - 120 (ln(n) + gamma) 2 2 + 1/6 (-720 (ln(n) + gamma) + 16320 ln(n) + 16320 gamma - 34684) Pi 6 4 88 Pi + 1/36 (8880 - 1440 ln(n) - 1440 gamma) Pi - ------ 7 2 2 + (1600 Pi + 7360 ln(n) + 7360 gamma - 25440) Zeta(3) + 2560 Zeta(3) 2 4 + 1/90 (960 Pi + 2880 ln(n) + 2880 gamma - 17760) Pi - 11520 Zeta(5) and in Maple format (74250517/2700-22600/9*Pi^2+140*Pi^4-88/7*Pi^6-6080*Zeta(3)+2560*Zeta(3)^2+1/90 *(960*Pi^2-10080)*Pi^4)*n^6+(174515669/900-22600/3*ln(n)-22600/3*gamma+1/6*(-\ 99080+5040*ln(n)+5040*gamma)*Pi^2+1/36*(29520-1440*ln(n)-1440*gamma)*Pi^4-528/7 *Pi^6+(1600*Pi^2-54240)*Zeta(3)+15360*Zeta(3)^2+1/90*(5760*Pi^2+2880*ln(n)+2880 *gamma-59040)*Pi^4-11520*Zeta(5))*n^5+(63183149/108-135140/3*ln(n)-135140/3* gamma+1260*(ln(n)+gamma)^2+1/6*(-866200/3+24480*ln(n)+24480*gamma-720*(ln(n)+ gamma)^2)*Pi^2+1/36*(76560-7200*ln(n)-7200*gamma)*Pi^4-1320/7*Pi^6+(8000*Pi^2+ 7360*ln(n)+7360*gamma-178240)*Zeta(3)+38400*Zeta(3)^2+1/90*(14400*Pi^2+14400*ln (n)+14400*gamma-153120)*Pi^4-57600*Zeta(5))*n^4+(514353749/540-128020*ln(n)-\ 128020*gamma+4860*(ln(n)+gamma)^2-120*(ln(n)+gamma)^3+1/6*(-1396040/3+59520*ln( n)+59520*gamma-2880*(ln(n)+gamma)^2)*Pi^2+1/36*(111840-14400*ln(n)-14400*gamma) *Pi^4-1760/7*Pi^6+(16000*Pi^2+29440*ln(n)+29440*gamma-295520)*Zeta(3)+51200* Zeta(3)^2+1/90*(19200*Pi^2+28800*ln(n)+28800*gamma-223680)*Pi^4-115200*Zeta(5)) *n^3+(390445043/450-562540/3*ln(n)-562540/3*gamma+12520*(ln(n)+gamma)^2-360*(ln (n)+gamma)^3+1/6*(-4320*(ln(n)+gamma)^2+82080*ln(n)+82080*gamma-420164)*Pi^2+1/ 36*(95760-14400*ln(n)-14400*gamma)*Pi^4-1320/7*Pi^6+(16000*Pi^2+44160*ln(n)+ 44160*gamma-268320)*Zeta(3)+38400*Zeta(3)^2+1/90*(14400*Pi^2+28800*ln(n)+28800* gamma-191520)*Pi^4-115200*Zeta(5))*n^2+(283347287/675-427874/3*ln(n)-427874/3* gamma+15500*(ln(n)+gamma)^2-360*(ln(n)+gamma)^3+1/6*(-582664/3+58320*ln(n)+ 58320*gamma-2880*(ln(n)+gamma)^2)*Pi^2+1/36*(44880-7200*ln(n)-7200*gamma)*Pi^4-\ 528/7*Pi^6+(8000*Pi^2+29440*ln(n)+29440*gamma-128320)*Zeta(3)+15360*Zeta(3)^2+1 /90*(5760*Pi^2+14400*ln(n)+14400*gamma-89760)*Pi^4-57600*Zeta(5))*n-45618*ln(n) -45618*gamma+6580*(ln(n)+gamma)^2-120*(ln(n)+gamma)^3+1/6*(-720*(ln(n)+gamma)^2 +16320*ln(n)+16320*gamma-34684)*Pi^2+1/36*(8880-1440*ln(n)-1440*gamma)*Pi^4-88/ 7*Pi^6+(1600*Pi^2+7360*ln(n)+7360*gamma-25440)*Zeta(3)+2560*Zeta(3)^2+1/90*(960 *Pi^2+2880*ln(n)+2880*gamma-17760)*Pi^4-11520*Zeta(5) and in LaTex \left( {\frac {74250517}{2700}}-{\frac {22600}{9}}\,{\pi }^{2}+140\,{ \pi }^{4}-{\frac {88}{7}}\,{\pi }^{6}-6080\,\zeta \left( 3 \right) + 2560\, \left( \zeta \left( 3 \right) \right) ^{2}+{\frac {1}{90}}\, \left( 960\,{\pi }^{2}-10080 \right) {\pi }^{4} \right) {n}^{6}+ \left( {\frac {174515669}{900}}-{\frac {22600}{3}}\,\ln \left( n \right) -{\frac {22600}{3}}\,\gamma+1/6\, \left( -99080+5040\,\ln \left( n \right) +5040\,\gamma \right) {\pi }^{2}+1/36\, \left( 29520- 1440\,\ln \left( n \right) -1440\,\gamma \right) {\pi }^{4}-{\frac { 528}{7}}\,{\pi }^{6}+ \left( 1600\,{\pi }^{2}-54240 \right) \zeta \left( 3 \right) +15360\, \left( \zeta \left( 3 \right) \right) ^{2} +{\frac {1}{90}}\, \left( 5760\,{\pi }^{2}+2880\,\ln \left( n \right) +2880\,\gamma-59040 \right) {\pi }^{4}-11520\,\zeta \left( 5 \right) \right) {n}^{5}+ \left( {\frac {63183149}{108}}-{\frac {135140}{3}}\, \ln \left( n \right) -{\frac {135140}{3}}\,\gamma+1260\, \left( \ln \left( n \right) +\gamma \right) ^{2}+1/6\, \left( -{\frac {866200}{3} }+24480\,\ln \left( n \right) +24480\,\gamma-720\, \left( \ln \left( n \right) +\gamma \right) ^{2} \right) {\pi }^{2}+1/36\, \left( 76560- 7200\,\ln \left( n \right) -7200\,\gamma \right) {\pi }^{4}-{\frac { 1320}{7}}\,{\pi }^{6}+ \left( 8000\,{\pi }^{2}+7360\,\ln \left( n \right) +7360\,\gamma-178240 \right) \zeta \left( 3 \right) +38400\, \left( \zeta \left( 3 \right) \right) ^{2}+{\frac {1}{90}}\, \left( 14400\,{\pi }^{2}+14400\,\ln \left( n \right) +14400\,\gamma-153120 \right) {\pi }^{4}-57600\,\zeta \left( 5 \right) \right) {n}^{4}+ \left( {\frac {514353749}{540}}-128020\,\ln \left( n \right) -128020 \,\gamma+4860\, \left( \ln \left( n \right) +\gamma \right) ^{2}-120\, \left( \ln \left( n \right) +\gamma \right) ^{3}+1/6\, \left( -{ \frac {1396040}{3}}+59520\,\ln \left( n \right) +59520\,\gamma-2880\, \left( \ln \left( n \right) +\gamma \right) ^{2} \right) {\pi }^{2}+1 /36\, \left( 111840-14400\,\ln \left( n \right) -14400\,\gamma \right) {\pi }^{4}-{\frac {1760}{7}}\,{\pi }^{6}+ \left( 16000\,{\pi } ^{2}+29440\,\ln \left( n \right) +29440\,\gamma-295520 \right) \zeta \left( 3 \right) +51200\, \left( \zeta \left( 3 \right) \right) ^{2} +{\frac {1}{90}}\, \left( 19200\,{\pi }^{2}+28800\,\ln \left( n \right) +28800\,\gamma-223680 \right) {\pi }^{4}-115200\,\zeta \left( 5 \right) \right) {n}^{3}+ \left( {\frac {390445043}{450}}-{ \frac {562540}{3}}\,\ln \left( n \right) -{\frac {562540}{3}}\,\gamma+ 12520\, \left( \ln \left( n \right) +\gamma \right) ^{2}-360\, \left( \ln \left( n \right) +\gamma \right) ^{3}+1/6\, \left( -4320\, \left( \ln \left( n \right) +\gamma \right) ^{2}+82080\,\ln \left( n \right) +82080\,\gamma-420164 \right) {\pi }^{2}+1/36\, \left( 95760- 14400\,\ln \left( n \right) -14400\,\gamma \right) {\pi }^{4}-{\frac { 1320}{7}}\,{\pi }^{6}+ \left( 16000\,{\pi }^{2}+44160\,\ln \left( n \right) +44160\,\gamma-268320 \right) \zeta \left( 3 \right) +38400\, \left( \zeta \left( 3 \right) \right) ^{2}+{\frac {1}{90}}\, \left( 14400\,{\pi }^{2}+28800\,\ln \left( n \right) +28800\,\gamma-191520 \right) {\pi }^{4}-115200\,\zeta \left( 5 \right) \right) {n}^{2}+ \left( {\frac {283347287}{675}}-{\frac {427874}{3}}\,\ln \left( n \right) -{\frac {427874}{3}}\,\gamma+15500\, \left( \ln \left( n \right) +\gamma \right) ^{2}-360\, \left( \ln \left( n \right) + \gamma \right) ^{3}+1/6\, \left( -{\frac {582664}{3}}+58320\,\ln \left( n \right) +58320\,\gamma-2880\, \left( \ln \left( n \right) + \gamma \right) ^{2} \right) {\pi }^{2}+1/36\, \left( 44880-7200\,\ln \left( n \right) -7200\,\gamma \right) {\pi }^{4}-{\frac {528}{7}}\,{ \pi }^{6}+ \left( 8000\,{\pi }^{2}+29440\,\ln \left( n \right) +29440 \,\gamma-128320 \right) \zeta \left( 3 \right) +15360\, \left( \zeta \left( 3 \right) \right) ^{2}+{\frac {1}{90}}\, \left( 5760\,{\pi }^{ 2}+14400\,\ln \left( n \right) +14400\,\gamma-89760 \right) {\pi }^{4} -57600\,\zeta \left( 5 \right) \right) n-45618\,\ln \left( n \right) -45618\,\gamma+6580\, \left( \ln \left( n \right) +\gamma \right) ^{2}-120\, \left( \ln \left( n \right) +\gamma \right) ^{3}+1 /6\, \left( -720\, \left( \ln \left( n \right) +\gamma \right) ^{2}+ 16320\,\ln \left( n \right) +16320\,\gamma-34684 \right) {\pi }^{2}+1/ 36\, \left( 8880-1440\,\ln \left( n \right) -1440\,\gamma \right) { \pi }^{4}-{\frac {88}{7}}\,{\pi }^{6}+ \left( 1600\,{\pi }^{2}+7360\, \ln \left( n \right) +7360\,\gamma-25440 \right) \zeta \left( 3 \right) +2560\, \left( \zeta \left( 3 \right) \right) ^{2}+{\frac {1 }{90}}\, \left( 960\,{\pi }^{2}+2880\,\ln \left( n \right) +2880\, \gamma-17760 \right) {\pi }^{4}-11520\,\zeta \left( 5 \right) In floating point this is 6 5 3.29771779470001265 n + (-67.322945016927745 - 22.1383646902915913 ln(n)) n + 2 (558.941859999476683 + 172.094455672432951 ln(n) + 75.6474718692769657 %1 ) 4 n + (-1395.47800848344266 - 2517.696394895301716 ln(n) 2 3 3 + 122.5898874771078627 %1 - 120. %1 ) n + (23.89010965252002 2 3 2 - 7207.03956562341876 ln(n) + 5413.8848312156617940 %1 - 360. %1 ) n + ( 7948.785970735906241 - 15199.920300419742620 ln(n) 2 3 + 10762.589887477107863 %1 - 360. %1 ) n - 10704.809950054350114 ln(n) 2 3 - 80102.144670211160876 + 5395.6474718692769657 %1 - 120. %1 %1 := ln(n) + 0.57721566490153286061 written in Maple format this is 3.29771779470001265*n^6+(-67.322945016927745-22.1383646902915913*ln(n))*n^5+( 558.941859999476683+172.094455672432951*ln(n)+75.6474718692769657*(ln(n)+.57721\ 566490153286061)^2)*n^4+(-1395.47800848344266-2517.696394895301716*ln(n)+122.58\ 98874771078627*(ln(n)+.57721566490153286061)^2-120.*(ln(n)+.5772156649015328606\ 1)^3)*n^3+(23.89010965252002-7207.03956562341876*ln(n)+5413.8848312156617940*( ln(n)+.57721566490153286061)^2-360.*(ln(n)+.57721566490153286061)^3)*n^2+(7948.\ 785970735906241-15199.920300419742620*ln(n)+10762.589887477107863*(ln(n)+.57721\ 566490153286061)^2-360.*(ln(n)+.57721566490153286061)^3)*n-10704.80995005435011\ 4*ln(n)-80102.144670211160876+5395.6474718692769657*(ln(n)+.5772156649015328606\ 1)^2-120.*(ln(n)+.57721566490153286061)^3 and in LaTex 3.29771779470001265*n^6+(-67.322945016927745-22.1383646902915913*ln(n))*n^5+( 558.941859999476683+172.094455672432951*ln(n)+75.6474718692769657*(ln(n)+.57721\ 566490153286061)^2)*n^4+(-1395.47800848344266-2517.696394895301716*ln(n)+122.58\ 98874771078627*(ln(n)+.57721566490153286061)^2-120.*(ln(n)+.5772156649015328606\ 1)^3)*n^3+(23.89010965252002-7207.03956562341876*ln(n)+5413.8848312156617940*( ln(n)+.57721566490153286061)^2-360.*(ln(n)+.57721566490153286061)^3)*n^2+(7948.\ 785970735906241-15199.920300419742620*ln(n)+10762.589887477107863*(ln(n)+.57721\ 566490153286061)^2-360.*(ln(n)+.57721566490153286061)^3)*n-10704.80995005435011\ 4*ln(n)-80102.144670211160876+5395.6474718692769657*(ln(n)+.5772156649015328606\ 1)^2-120.*(ln(n)+.57721566490153286061)^3 The limit of the scaled , 6, -th moment is / 2 6 |74250517 22600 Pi 4 88 Pi 2 |-------- - --------- + 140 Pi - ------ - 6080 Zeta(3) + 2560 Zeta(3) \ 2700 9 7 2 4\ / 2\3 (960 Pi - 10080) Pi | / | 2 Pi | + ---------------------| / |7 - -----| 90 / / \ 3 / in Maple format this is: (74250517/2700-22600/9*Pi^2+140*Pi^4-88/7*Pi^6-6080*Zeta(3)+2560*Zeta(3)^2+1/90 *(960*Pi^2-10080)*Pi^4)/(7-2/3*Pi^2)^3 in LaTex this is: {\frac {{\frac {74250517}{2700}}-{\frac {22600}{9}}\,{\pi }^{2}+140\,{ \pi }^{4}-{\frac {88}{7}}\,{\pi }^{6}-6080\,\zeta \left( 3 \right) + 2560\, \left( \zeta \left( 3 \right) \right) ^{2}+{\frac {1}{90}}\, \left( 960\,{\pi }^{2}-10080 \right) {\pi }^{4}}{ \left( 7-2/3\,{\pi } ^{2} \right) ^{3}}} In floating point, this is , 6, -th moment is 44.427077708169777614 in Maple format this is: 44.427077708169777614 in LaTex this is: 44.427077708169777614 Theorem No. , 7, : The , 7, -th moment about the mean of the random variabl\ e "number of comparisons in Quicksort applied to lists of length n" is 6 5 4 3 - 7 n (4361821529 n + 36987323153 n + 135379990924 n + 280732243290 n 2 + 356835588161 n + 275702876557 n + 119672624386)/81000 + 1/18 (n + 1) ( 5 4 3 2 1607347 n + 12169045 n + 35117495 n + 56653667 n + 48253086 n + 19517436) H[1](n) 3 2 2 2 - 28 (285 n + 2685 n + 4290 n + 6208) (n + 1) H[1](n) 3 3 + 5880 (n + 1) H[1](n) + (7 5 4 3 2 (229621 n + 1557635 n + 4086825 n + 5641645 n + 4118354 n + 1044612) 2 3 2 3 (n + 1) /9 - 112 (285 n + 2265 n + 3510 n + 3559) (n + 1) H[1](n) 4 2 + 28560 (n + 1) H[1](n) ) H[2](n) + 3 2 4 5 (-1680 (19 n + 123 n + 182 n + 122) (n + 1) + 43680 (n + 1) H[1](n)) / 2 6 3 | H[2](n) + 20160 (n + 1) H[2](n) + | \ 4 3 2 3 112 (11300 n + 53420 n + 115420 n + 114700 n + 47367) (n + 1) ----------------------------------------------------------------- 9 2 4 5 2 - 2240 (21 n + 39 n + 85) (n + 1) H[1](n) + 6720 (n + 1) H[1](n) 2 5 6 + (-4480 (21 n + 39 n + 50) (n + 1) + 26880 (n + 1) H[1](n)) H[2](n) \ 7 2| 6 2 + 26880 (n + 1) H[2](n) | H[3](n) - 53760 (n + 1) H[3](n) + ( / 3 2 4 5 3360 (19 n + 123 n + 182 n + 122) (n + 1) - 87360 (n + 1) H[1](n) 6 7 - 120960 (n + 1) H[2](n) - 53760 (n + 1) H[3](n)) H[4](n) + ( 2 5 6 5376 (21 n + 39 n + 50) (n + 1) - 32256 (n + 1) H[1](n) 7 6 - 64512 (n + 1) H[2](n)) H[5](n) + 161280 (n + 1) H[6](n) 7 + 92160 (n + 1) H[7](n) and in Maple format -7/81000*n*(4361821529*n^6+36987323153*n^5+135379990924*n^4+280732243290*n^3+ 356835588161*n^2+275702876557*n+119672624386)+1/18*(n+1)*(1607347*n^5+12169045* n^4+35117495*n^3+56653667*n^2+48253086*n+19517436)*H[1](n)-28*(285*n^3+2685*n^2 +4290*n+6208)*(n+1)^2*H[1](n)^2+5880*(n+1)^3*H[1](n)^3+(7/9*(229621*n^5+1557635 *n^4+4086825*n^3+5641645*n^2+4118354*n+1044612)*(n+1)^2-112*(285*n^3+2265*n^2+ 3510*n+3559)*(n+1)^3*H[1](n)+28560*(n+1)^4*H[1](n)^2)*H[2](n)+(-1680*(19*n^3+ 123*n^2+182*n+122)*(n+1)^4+43680*(n+1)^5*H[1](n))*H[2](n)^2+20160*(n+1)^6*H[2]( n)^3+(112/9*(11300*n^4+53420*n^3+115420*n^2+114700*n+47367)*(n+1)^3-2240*(21*n^ 2+39*n+85)*(n+1)^4*H[1](n)+6720*(n+1)^5*H[1](n)^2+(-4480*(21*n^2+39*n+50)*(n+1) ^5+26880*(n+1)^6*H[1](n))*H[2](n)+26880*(n+1)^7*H[2](n)^2)*H[3](n)-53760*(n+1)^ 6*H[3](n)^2+(3360*(19*n^3+123*n^2+182*n+122)*(n+1)^4-87360*(n+1)^5*H[1](n)-\ 120960*(n+1)^6*H[2](n)-53760*(n+1)^7*H[3](n))*H[4](n)+(5376*(21*n^2+39*n+50)*(n +1)^5-32256*(n+1)^6*H[1](n)-64512*(n+1)^7*H[2](n))*H[5](n)+161280*(n+1)^6*H[6]( n)+92160*(n+1)^7*H[7](n) and in LaTex -{\frac {7}{81000}}\,n \left( 4361821529\,{n}^{6}+36987323153\,{n}^{5}+ 135379990924\,{n}^{4}+280732243290\,{n}^{3}+356835588161\,{n}^{2}+ 275702876557\,n+119672624386 \right) +1/18\, \left( n+1 \right) \left( 1607347\,{n}^{5}+12169045\,{n}^{4}+35117495\,{n}^{3}+56653667\, {n}^{2}+48253086\,n+19517436 \right) H_{{1}} \left( n \right) -28\, \left( 285\,{n}^{3}+2685\,{n}^{2}+4290\,n+6208 \right) \left( n+1 \right) ^{2} \left( H_{{1}} \left( n \right) \right) ^{2}+5880\, \left( n+1 \right) ^{3} \left( H_{{1}} \left( n \right) \right) ^{3}+ \left( {\frac {7}{9}}\, \left( 229621\,{n}^{5}+1557635\,{n}^{4}+ 4086825\,{n}^{3}+5641645\,{n}^{2}+4118354\,n+1044612 \right) \left( n+ 1 \right) ^{2}-112\, \left( 285\,{n}^{3}+2265\,{n}^{2}+3510\,n+3559 \right) \left( n+1 \right) ^{3}H_{{1}} \left( n \right) +28560\, \left( n+1 \right) ^{4} \left( H_{{1}} \left( n \right) \right) ^{2} \right) H_{{2}} \left( n \right) + \left( -1680\, \left( 19\,{n}^{3}+ 123\,{n}^{2}+182\,n+122 \right) \left( n+1 \right) ^{4}+43680\, \left( n+1 \right) ^{5}H_{{1}} \left( n \right) \right) \left( H_{{2 }} \left( n \right) \right) ^{2}+20160\, \left( n+1 \right) ^{6} \left( H_{{2}} \left( n \right) \right) ^{3}+ \left( {\frac {112}{9}} \, \left( 11300\,{n}^{4}+53420\,{n}^{3}+115420\,{n}^{2}+114700\,n+47367 \right) \left( n+1 \right) ^{3}-2240\, \left( 21\,{n}^{2}+39\,n+85 \right) \left( n+1 \right) ^{4}H_{{1}} \left( n \right) +6720\, \left( n+1 \right) ^{5} \left( H_{{1}} \left( n \right) \right) ^{2}+ \left( -4480\, \left( 21\,{n}^{2}+39\,n+50 \right) \left( n+1 \right) ^{5}+26880\, \left( n+1 \right) ^{6}H_{{1}} \left( n \right) \right) H_{{2}} \left( n \right) +26880\, \left( n+1 \right) ^{7} \left( H_{{2}} \left( n \right) \right) ^{2} \right) H_{{3}} \left( n \right) -53760\, \left( n+1 \right) ^{6} \left( H_{{3}} \left( n \right) \right) ^{2}+ \left( 3360\, \left( 19\,{n}^{3}+123\,{n}^{2}+ 182\,n+122 \right) \left( n+1 \right) ^{4}-87360\, \left( n+1 \right) ^{5}H_{{1}} \left( n \right) -120960\, \left( n+1 \right) ^{6}H_{{2}} \left( n \right) -53760\, \left( n+1 \right) ^{7}H_{{3}} \left( n \right) \right) H_{{4}} \left( n \right) + \left( 5376\, \left( 21\,{ n}^{2}+39\,n+50 \right) \left( n+1 \right) ^{5}-32256\, \left( n+1 \right) ^{6}H_{{1}} \left( n \right) -64512\, \left( n+1 \right) ^{7}H _{{2}} \left( n \right) \right) H_{{5}} \left( n \right) +161280\, \left( n+1 \right) ^{6}H_{{6}} \left( n \right) +92160\, \left( n+1 \right) ^{7}H_{{7}} \left( n \right) This is asymptotically / 2 4 | 30532750703 1607347 Pi 2660 Pi |- ----------- + ----------- - -------- \ 81000 54 3 2 4 + (1265600/9 - 15680 Pi + 2240/3 Pi ) Zeta(3) 4 2 + 1/90 (63840 - 53760 Zeta(3)) Pi + (-10752 Pi + 112896) Zeta(5) \ / | 7 | 258911262071 1607347 1607347 gamma + 92160 Zeta(7)| n + |- ------------ + ------- ln(n) + ------------- / \ 81000 18 18 4 2 27860 Pi 6 + 1/6 (14118139/9 - 31920 ln(n) - 31920 gamma) Pi - --------- + 264 Pi 3 / | + |9779840/9 - 47040 ln(n) - 47040 gamma \ 4\ 2 15680 Pi | + 1/6 (-645120 + 26880 ln(n) + 26880 gamma) Pi + ---------| Zeta(3) 3 / 2 2 4 - 53760 Zeta(3) + 1/90 (668640 - 20160 Pi - 376320 Zeta(3)) Pi 2 + (-75264 Pi - 32256 ln(n) - 32256 gamma + 774144) Zeta(5) \ | 6 / 236914984117 6888196 gamma + 645120 Zeta(7)| n + |- ------------ + 6888196/9 ln(n) + ------------- / \ 20250 9 2 - 7980 (ln(n) + gamma) 2 + 1/6 (52022012/9 - 349440 ln(n) - 349440 gamma) Pi 4 6 + 1/36 (-1323840 + 43680 ln(n) + 43680 gamma) Pi + 1584 Pi + (34672960/9 2 - 275520 ln(n) - 275520 gamma + 6720 (ln(n) + gamma) 2 4 + 1/6 (-2038400 + 161280 ln(n) + 161280 gamma) Pi + 15680 Pi ) Zeta(3) 2 - 322560 Zeta(3) + 1/90 2 4 (2647680 - 87360 ln(n) - 87360 gamma - 120960 Pi - 1128960 Zeta(3)) Pi 2 + (-225792 Pi - 193536 ln(n) - 193536 gamma + 2446080) Zeta(5) / \ 5 | 65504190101 + 1935360 Zeta(7)| n + |- ----------- + 2627030 ln(n) + 2627030 gamma / \ 2700 2 - 91140 (ln(n) + gamma) + 1/6 2 2 (35870170/3 - 1249920 ln(n) - 1249920 gamma + 28560 (ln(n) + gamma) ) Pi / 4 6 | + 1/36 (-2795520 + 218400 ln(n) + 218400 gamma) Pi + 3960 Pi + |7871360 \ 2 - 822080 ln(n) - 822080 gamma + 33600 (ln(n) + gamma) 4\ 2 78400 Pi | + 1/6 (-3808000 + 403200 ln(n) + 403200 gamma) Pi + ---------| Zeta(3) 3 / 2 - 806400 Zeta(3) + 1/90 2 4 (5591040 - 436800 ln(n) - 436800 gamma - 302400 Pi - 1881600 Zeta(3)) Pi 2 + (-376320 Pi - 483840 ln(n) - 483840 gamma + 4569600) Zeta(5) \ / | 4 | 2497849117127 + 3225600 Zeta(7)| n + |- ------------- + 45885581/9 ln(n) / \ 81000 45885581 gamma 2 3 + -------------- - 278460 (ln(n) + gamma) + 5880 (ln(n) + gamma) + 1/6 9 2 2 (136419283/9 - 2370928 ln(n) - 2370928 gamma + 114240 (ln(n) + gamma) ) Pi / 4 6 | + 1/36 (-3512880 + 436800 ln(n) + 436800 gamma) Pi + 5280 Pi + | \ 2 88608464/9 - 1473920 ln(n) - 1473920 gamma + 67200 (ln(n) + gamma) 4\ 2 78400 Pi | + 1/6 (-4457600 + 537600 ln(n) + 537600 gamma) Pi + ---------| Zeta(3) 3 / 2 - 1075200 Zeta(3) + 1/90 2 4 (7025760 - 873600 ln(n) - 873600 gamma - 403200 Pi - 1881600 Zeta(3)) Pi 2 + (-376320 Pi - 645120 ln(n) - 645120 gamma + 5349120) Zeta(5) \ | 3 / 1929920135899 104906753 + 3225600 Zeta(7)| n + |- ------------- + --------- ln(n) / \ 81000 18 104906753 gamma 2 3 + --------------- - 489244 (ln(n) + gamma) + 17640 (ln(n) + gamma) + 1/6 18 2 2 (104460755/9 - 2628864 ln(n) - 2628864 gamma + 171360 (ln(n) + gamma) ) Pi 4 6 + 1/36 (-2659440 + 436800 ln(n) + 436800 gamma) Pi + 3960 Pi + ( 2 67381552/9 - 1538880 ln(n) - 1538880 gamma + 67200 (ln(n) + gamma) 2 4 + 1/6 (-3207680 + 403200 ln(n) + 403200 gamma) Pi + 15680 Pi ) Zeta(3) 2 - 806400 Zeta(3) + 1/90 2 4 (5318880 - 873600 ln(n) - 873600 gamma - 302400 Pi - 1128960 Zeta(3)) Pi 2 + (-225792 Pi - 483840 ln(n) - 483840 gamma + 3849216) Zeta(5) / \ 2 | 418854185351 + 1935360 Zeta(7)| n + |- ------------ + 3765029 ln(n) + 3765029 gamma / \ 40500 2 3 - 467768 (ln(n) + gamma) + 17640 (ln(n) + gamma) + 1/6 2 2 (43453046/9 - 1588944 ln(n) - 1588944 gamma + 114240 (ln(n) + gamma) ) Pi / 4 6 | + 1/36 (-1125600 + 218400 ln(n) + 218400 gamma) Pi + 1584 Pi + | \ 2 28761712/9 - 848960 ln(n) - 848960 gamma + 33600 (ln(n) + gamma) 4\ 2 15680 Pi | + 1/6 (-1294720 + 161280 ln(n) + 161280 gamma) Pi + ---------| Zeta(3) 3 / 2 - 322560 Zeta(3) + 1/90 2 4 (2251200 - 436800 ln(n) - 436800 gamma - 120960 Pi - 376320 Zeta(3)) Pi 2 + (-75264 Pi - 193536 ln(n) - 193536 gamma + 1553664) Zeta(5) \ | + 645120 Zeta(7)| n + 1084302 ln(n) + 1084302 gamma / 2 3 - 173824 (ln(n) + gamma) + 5880 (ln(n) + gamma) 2 2 + 1/6 (28560 (ln(n) + gamma) - 398608 ln(n) - 398608 gamma + 812476) Pi / 4 6 | + 1/36 (-204960 + 43680 ln(n) + 43680 gamma) Pi + 264 Pi + |589456 \ 2 - 190400 ln(n) - 190400 gamma + 6720 (ln(n) + gamma) 4\ 2 2240 Pi | + 1/6 (-224000 + 26880 ln(n) + 26880 gamma) Pi + --------| Zeta(3) 3 / 2 - 53760 Zeta(3) + 2 4 1/90 (409920 - 87360 ln(n) - 87360 gamma - 20160 Pi - 53760 Zeta(3)) Pi 2 + (-10752 Pi - 32256 ln(n) - 32256 gamma + 268800) Zeta(5) + 92160 Zeta(7) and in Maple format (-30532750703/81000+1607347/54*Pi^2-2660/3*Pi^4+(1265600/9-15680*Pi^2+2240/3*Pi ^4)*Zeta(3)+1/90*(63840-53760*Zeta(3))*Pi^4+(-10752*Pi^2+112896)*Zeta(5)+92160* Zeta(7))*n^7+(-258911262071/81000+1607347/18*ln(n)+1607347/18*gamma+1/6*( 14118139/9-31920*ln(n)-31920*gamma)*Pi^2-27860/3*Pi^4+264*Pi^6+(9779840/9-47040 *ln(n)-47040*gamma+1/6*(-645120+26880*ln(n)+26880*gamma)*Pi^2+15680/3*Pi^4)* Zeta(3)-53760*Zeta(3)^2+1/90*(668640-20160*Pi^2-376320*Zeta(3))*Pi^4+(-75264*Pi ^2-32256*ln(n)-32256*gamma+774144)*Zeta(5)+645120*Zeta(7))*n^6+(-236914984117/ 20250+6888196/9*ln(n)+6888196/9*gamma-7980*(ln(n)+gamma)^2+1/6*(52022012/9-\ 349440*ln(n)-349440*gamma)*Pi^2+1/36*(-1323840+43680*ln(n)+43680*gamma)*Pi^4+ 1584*Pi^6+(34672960/9-275520*ln(n)-275520*gamma+6720*(ln(n)+gamma)^2+1/6*(-\ 2038400+161280*ln(n)+161280*gamma)*Pi^2+15680*Pi^4)*Zeta(3)-322560*Zeta(3)^2+1/ 90*(2647680-87360*ln(n)-87360*gamma-120960*Pi^2-1128960*Zeta(3))*Pi^4+(-225792* Pi^2-193536*ln(n)-193536*gamma+2446080)*Zeta(5)+1935360*Zeta(7))*n^5+(-\ 65504190101/2700+2627030*ln(n)+2627030*gamma-91140*(ln(n)+gamma)^2+1/6*( 35870170/3-1249920*ln(n)-1249920*gamma+28560*(ln(n)+gamma)^2)*Pi^2+1/36*(-\ 2795520+218400*ln(n)+218400*gamma)*Pi^4+3960*Pi^6+(7871360-822080*ln(n)-822080* gamma+33600*(ln(n)+gamma)^2+1/6*(-3808000+403200*ln(n)+403200*gamma)*Pi^2+78400 /3*Pi^4)*Zeta(3)-806400*Zeta(3)^2+1/90*(5591040-436800*ln(n)-436800*gamma-\ 302400*Pi^2-1881600*Zeta(3))*Pi^4+(-376320*Pi^2-483840*ln(n)-483840*gamma+ 4569600)*Zeta(5)+3225600*Zeta(7))*n^4+(-2497849117127/81000+45885581/9*ln(n)+ 45885581/9*gamma-278460*(ln(n)+gamma)^2+5880*(ln(n)+gamma)^3+1/6*(136419283/9-\ 2370928*ln(n)-2370928*gamma+114240*(ln(n)+gamma)^2)*Pi^2+1/36*(-3512880+436800* ln(n)+436800*gamma)*Pi^4+5280*Pi^6+(88608464/9-1473920*ln(n)-1473920*gamma+ 67200*(ln(n)+gamma)^2+1/6*(-4457600+537600*ln(n)+537600*gamma)*Pi^2+78400/3*Pi^ 4)*Zeta(3)-1075200*Zeta(3)^2+1/90*(7025760-873600*ln(n)-873600*gamma-403200*Pi^ 2-1881600*Zeta(3))*Pi^4+(-376320*Pi^2-645120*ln(n)-645120*gamma+5349120)*Zeta(5 )+3225600*Zeta(7))*n^3+(-1929920135899/81000+104906753/18*ln(n)+104906753/18* gamma-489244*(ln(n)+gamma)^2+17640*(ln(n)+gamma)^3+1/6*(104460755/9-2628864*ln( n)-2628864*gamma+171360*(ln(n)+gamma)^2)*Pi^2+1/36*(-2659440+436800*ln(n)+ 436800*gamma)*Pi^4+3960*Pi^6+(67381552/9-1538880*ln(n)-1538880*gamma+67200*(ln( n)+gamma)^2+1/6*(-3207680+403200*ln(n)+403200*gamma)*Pi^2+15680*Pi^4)*Zeta(3)-\ 806400*Zeta(3)^2+1/90*(5318880-873600*ln(n)-873600*gamma-302400*Pi^2-1128960* Zeta(3))*Pi^4+(-225792*Pi^2-483840*ln(n)-483840*gamma+3849216)*Zeta(5)+1935360* Zeta(7))*n^2+(-418854185351/40500+3765029*ln(n)+3765029*gamma-467768*(ln(n)+ gamma)^2+17640*(ln(n)+gamma)^3+1/6*(43453046/9-1588944*ln(n)-1588944*gamma+ 114240*(ln(n)+gamma)^2)*Pi^2+1/36*(-1125600+218400*ln(n)+218400*gamma)*Pi^4+ 1584*Pi^6+(28761712/9-848960*ln(n)-848960*gamma+33600*(ln(n)+gamma)^2+1/6*(-\ 1294720+161280*ln(n)+161280*gamma)*Pi^2+15680/3*Pi^4)*Zeta(3)-322560*Zeta(3)^2+ 1/90*(2251200-436800*ln(n)-436800*gamma-120960*Pi^2-376320*Zeta(3))*Pi^4+(-\ 75264*Pi^2-193536*ln(n)-193536*gamma+1553664)*Zeta(5)+645120*Zeta(7))*n+1084302 *ln(n)+1084302*gamma-173824*(ln(n)+gamma)^2+5880*(ln(n)+gamma)^3+1/6*(28560*(ln (n)+gamma)^2-398608*ln(n)-398608*gamma+812476)*Pi^2+1/36*(-204960+43680*ln(n)+ 43680*gamma)*Pi^4+264*Pi^6+(589456-190400*ln(n)-190400*gamma+6720*(ln(n)+gamma) ^2+1/6*(-224000+26880*ln(n)+26880*gamma)*Pi^2+2240/3*Pi^4)*Zeta(3)-53760*Zeta(3 )^2+1/90*(409920-87360*ln(n)-87360*gamma-20160*Pi^2-53760*Zeta(3))*Pi^4+(-10752 *Pi^2-32256*ln(n)-32256*gamma+268800)*Zeta(5)+92160*Zeta(7) and in LaTex \left( -{\frac {30532750703}{81000}}+{\frac {1607347}{54}}\,{\pi }^{2} -{\frac {2660}{3}}\,{\pi }^{4}+ \left( {\frac {1265600}{9}}-15680\,{ \pi }^{2}+{\frac {2240}{3}}\,{\pi }^{4} \right) \zeta \left( 3 \right) +{\frac {1}{90}}\, \left( 63840-53760\,\zeta \left( 3 \right) \right) {\pi }^{4}+ \left( -10752\,{\pi }^{2}+112896 \right) \zeta \left( 5 \right) +92160\,\zeta \left( 7 \right) \right) {n}^{7 }+ \left( -{\frac {258911262071}{81000}}+{\frac {1607347}{18}}\,\ln \left( n \right) +{\frac {1607347}{18}}\,\gamma+1/6\, \left( {\frac { 14118139}{9}}-31920\,\ln \left( n \right) -31920\,\gamma \right) {\pi }^{2}-{\frac {27860}{3}}\,{\pi }^{4}+264\,{\pi }^{6}+ \left( {\frac { 9779840}{9}}-47040\,\ln \left( n \right) -47040\,\gamma+1/6\, \left( - 645120+26880\,\ln \left( n \right) +26880\,\gamma \right) {\pi }^{2}+{ \frac {15680}{3}}\,{\pi }^{4} \right) \zeta \left( 3 \right) -53760\, \left( \zeta \left( 3 \right) \right) ^{2}+{\frac {1}{90}}\, \left( 668640-20160\,{\pi }^{2}-376320\,\zeta \left( 3 \right) \right) {\pi }^{4}+ \left( -75264\,{\pi }^{2}-32256\,\ln \left( n \right) -32256\, \gamma+774144 \right) \zeta \left( 5 \right) +645120\,\zeta \left( 7 \right) \right) {n}^{6}+ \left( -{\frac {236914984117}{20250}}+{ \frac {6888196}{9}}\,\ln \left( n \right) +{\frac {6888196}{9}}\, \gamma-7980\, \left( \ln \left( n \right) +\gamma \right) ^{2}+1/6\, \left( {\frac {52022012}{9}}-349440\,\ln \left( n \right) -349440\, \gamma \right) {\pi }^{2}+1/36\, \left( -1323840+43680\,\ln \left( n \right) +43680\,\gamma \right) {\pi }^{4}+1584\,{\pi }^{6}+ \left( { \frac {34672960}{9}}-275520\,\ln \left( n \right) -275520\,\gamma+6720 \, \left( \ln \left( n \right) +\gamma \right) ^{2}+1/6\, \left( - 2038400+161280\,\ln \left( n \right) +161280\,\gamma \right) {\pi }^{2 }+15680\,{\pi }^{4} \right) \zeta \left( 3 \right) -322560\, \left( \zeta \left( 3 \right) \right) ^{2}+{\frac {1}{90}}\, \left( 2647680- 87360\,\ln \left( n \right) -87360\,\gamma-120960\,{\pi }^{2}-1128960 \,\zeta \left( 3 \right) \right) {\pi }^{4}+ \left( -225792\,{\pi }^{ 2}-193536\,\ln \left( n \right) -193536\,\gamma+2446080 \right) \zeta \left( 5 \right) +1935360\,\zeta \left( 7 \right) \right) {n}^{5}+ \left( -{\frac {65504190101}{2700}}+2627030\,\ln \left( n \right) + 2627030\,\gamma-91140\, \left( \ln \left( n \right) +\gamma \right) ^{ 2}+1/6\, \left( {\frac {35870170}{3}}-1249920\,\ln \left( n \right) - 1249920\,\gamma+28560\, \left( \ln \left( n \right) +\gamma \right) ^{ 2} \right) {\pi }^{2}+1/36\, \left( -2795520+218400\,\ln \left( n \right) +218400\,\gamma \right) {\pi }^{4}+3960\,{\pi }^{6}+ \left( 7871360-822080\,\ln \left( n \right) -822080\,\gamma+33600\, \left( \ln \left( n \right) +\gamma \right) ^{2}+1/6\, \left( -3808000+403200 \,\ln \left( n \right) +403200\,\gamma \right) {\pi }^{2}+{\frac { 78400}{3}}\,{\pi }^{4} \right) \zeta \left( 3 \right) -806400\, \left( \zeta \left( 3 \right) \right) ^{2}+{\frac {1}{90}}\, \left( 5591040-436800\,\ln \left( n \right) -436800\,\gamma-302400\,{\pi }^{2 }-1881600\,\zeta \left( 3 \right) \right) {\pi }^{4}+ \left( -376320 \,{\pi }^{2}-483840\,\ln \left( n \right) -483840\,\gamma+4569600 \right) \zeta \left( 5 \right) +3225600\,\zeta \left( 7 \right) \right) {n}^{4}+ \left( -{\frac {2497849117127}{81000}}+{\frac { 45885581}{9}}\,\ln \left( n \right) +{\frac {45885581}{9}}\,\gamma- 278460\, \left( \ln \left( n \right) +\gamma \right) ^{2}+5880\, \left( \ln \left( n \right) +\gamma \right) ^{3}+1/6\, \left( {\frac {136419283}{9}}-2370928\,\ln \left( n \right) -2370928\,\gamma+114240 \, \left( \ln \left( n \right) +\gamma \right) ^{2} \right) {\pi }^{2} +1/36\, \left( -3512880+436800\,\ln \left( n \right) +436800\,\gamma \right) {\pi }^{4}+5280\,{\pi }^{6}+ \left( {\frac {88608464}{9}}- 1473920\,\ln \left( n \right) -1473920\,\gamma+67200\, \left( \ln \left( n \right) +\gamma \right) ^{2}+1/6\, \left( -4457600+537600\, \ln \left( n \right) +537600\,\gamma \right) {\pi }^{2}+{\frac {78400} {3}}\,{\pi }^{4} \right) \zeta \left( 3 \right) -1075200\, \left( \zeta \left( 3 \right) \right) ^{2}+{\frac {1}{90}}\, \left( 7025760- 873600\,\ln \left( n \right) -873600\,\gamma-403200\,{\pi }^{2}- 1881600\,\zeta \left( 3 \right) \right) {\pi }^{4}+ \left( -376320\,{ \pi }^{2}-645120\,\ln \left( n \right) -645120\,\gamma+5349120 \right) \zeta \left( 5 \right) +3225600\,\zeta \left( 7 \right) \right) {n}^{3}+ \left( -{\frac {1929920135899}{81000}}+{\frac { 104906753}{18}}\,\ln \left( n \right) +{\frac {104906753}{18}}\,\gamma -489244\, \left( \ln \left( n \right) +\gamma \right) ^{2}+17640\, \left( \ln \left( n \right) +\gamma \right) ^{3}+1/6\, \left( {\frac {104460755}{9}}-2628864\,\ln \left( n \right) -2628864\,\gamma+171360 \, \left( \ln \left( n \right) +\gamma \right) ^{2} \right) {\pi }^{2} +1/36\, \left( -2659440+436800\,\ln \left( n \right) +436800\,\gamma \right) {\pi }^{4}+3960\,{\pi }^{6}+ \left( {\frac {67381552}{9}}- 1538880\,\ln \left( n \right) -1538880\,\gamma+67200\, \left( \ln \left( n \right) +\gamma \right) ^{2}+1/6\, \left( -3207680+403200\, \ln \left( n \right) +403200\,\gamma \right) {\pi }^{2}+15680\,{\pi }^ {4} \right) \zeta \left( 3 \right) -806400\, \left( \zeta \left( 3 \right) \right) ^{2}+{\frac {1}{90}}\, \left( 5318880-873600\,\ln \left( n \right) -873600\,\gamma-302400\,{\pi }^{2}-1128960\,\zeta \left( 3 \right) \right) {\pi }^{4}+ \left( -225792\,{\pi }^{2}- 483840\,\ln \left( n \right) -483840\,\gamma+3849216 \right) \zeta \left( 5 \right) +1935360\,\zeta \left( 7 \right) \right) {n}^{2}+ \left( -{\frac {418854185351}{40500}}+3765029\,\ln \left( n \right) + 3765029\,\gamma-467768\, \left( \ln \left( n \right) +\gamma \right) ^ {2}+17640\, \left( \ln \left( n \right) +\gamma \right) ^{3}+1/6\, \left( {\frac {43453046}{9}}-1588944\,\ln \left( n \right) -1588944\, \gamma+114240\, \left( \ln \left( n \right) +\gamma \right) ^{2} \right) {\pi }^{2}+1/36\, \left( -1125600+218400\,\ln \left( n \right) +218400\,\gamma \right) {\pi }^{4}+1584\,{\pi }^{6}+ \left( { \frac {28761712}{9}}-848960\,\ln \left( n \right) -848960\,\gamma+ 33600\, \left( \ln \left( n \right) +\gamma \right) ^{2}+1/6\, \left( -1294720+161280\,\ln \left( n \right) +161280\,\gamma \right) {\pi }^{ 2}+{\frac {15680}{3}}\,{\pi }^{4} \right) \zeta \left( 3 \right) - 322560\, \left( \zeta \left( 3 \right) \right) ^{2}+{\frac {1}{90}}\, \left( 2251200-436800\,\ln \left( n \right) -436800\,\gamma-120960\,{ \pi }^{2}-376320\,\zeta \left( 3 \right) \right) {\pi }^{4}+ \left( - 75264\,{\pi }^{2}-193536\,\ln \left( n \right) -193536\,\gamma+1553664 \right) \zeta \left( 5 \right) +645120\,\zeta \left( 7 \right) \right) n+1084302\,\ln \left( n \right) +1084302\,\gamma-173824\, \left( \ln \left( n \right) +\gamma \right) ^{2}+5880\, \left( \ln \left( n \right) +\gamma \right) ^{3}+1/6\, \left( 28560\, \left( \ln \left( n \right) +\gamma \right) ^{2}-398608\,\ln \left( n \right) - 398608\,\gamma+812476 \right) {\pi }^{2}+1/36\, \left( -204960+43680\, \ln \left( n \right) +43680\,\gamma \right) {\pi }^{4}+264\,{\pi }^{6} + \left( 589456-190400\,\ln \left( n \right) -190400\,\gamma+6720\, \left( \ln \left( n \right) +\gamma \right) ^{2}+1/6\, \left( -224000 +26880\,\ln \left( n \right) +26880\,\gamma \right) {\pi }^{2}+{\frac {2240}{3}}\,{\pi }^{4} \right) \zeta \left( 3 \right) -53760\, \left( \zeta \left( 3 \right) \right) ^{2}+{\frac {1}{90}}\, \left( 409920- 87360\,\ln \left( n \right) -87360\,\gamma-20160\,{\pi }^{2}-53760\, \zeta \left( 3 \right) \right) {\pi }^{4}+ \left( -10752\,{\pi }^{2}- 32256\,\ln \left( n \right) -32256\,\gamma+268800 \right) \zeta \left( 5 \right) +92160\,\zeta \left( 7 \right) In floating point this is 2 4 (-12145.1555983825982 - 3461.2416718164774 ln(n) - 3771.571104652284983 %1 ) n 7 + 0.18713903445849370167 10 + 2 (2326.9329247408422 + 1213.36785390738162 ln(n) + 97.8223892324735979 %1 ) 5 n + (2294.2871959552956 + 185945.0498331263185 ln(n) 2 3 2 - 126589.87441256318185 %1 + 17640. %1 ) n + (23768.1541687017570 2 3 3 + 57077.3205038160383 ln(n) - 9764.50831093387592 %1 + 5880. %1 ) n + ( -137554.59122817500533 + 367229.1482973194947 ln(n) 2 3 7 - 239461.62025709624391 %1 + 17640. %1 ) n + 8.648240250323651 n 6 + (-236.33207699179703 - 51.197316780368167 ln(n)) n 3 2 + 243087.22581051344207 ln(n) + 5880. %1 - 118766.86066158217937 %1 %1 := ln(n) + 0.57721566490153286061 written in Maple format this is (-12145.1555983825982-3461.2416718164774*ln(n)-3771.571104652284983*(ln(n)+.577\ 21566490153286061)^2)*n^4+1871390.3445849370167+(2326.9329247408422+1213.367853\ 90738162*ln(n)+97.8223892324735979*(ln(n)+.57721566490153286061)^2)*n^5+(2294.2\ 871959552956+185945.0498331263185*ln(n)-126589.87441256318185*(ln(n)+.577215664\ 90153286061)^2+17640.*(ln(n)+.57721566490153286061)^3)*n^2+(23768.1541687017570 +57077.3205038160383*ln(n)-9764.50831093387592*(ln(n)+.57721566490153286061)^2+ 5880.*(ln(n)+.57721566490153286061)^3)*n^3+(-137554.59122817500533+367229.14829\ 73194947*ln(n)-239461.62025709624391*(ln(n)+.57721566490153286061)^2+17640.*(ln (n)+.57721566490153286061)^3)*n+8.648240250323651*n^7+(-236.33207699179703-51.1\ 97316780368167*ln(n))*n^6+243087.22581051344207*ln(n)+5880.*(ln(n)+.57721566490\ 153286061)^3-118766.86066158217937*(ln(n)+.57721566490153286061)^2 and in LaTex (-12145.1555983825982-3461.2416718164774*ln(n)-3771.571104652284983*(ln(n)+.577\ 21566490153286061)^2)*n^4+1871390.3445849370167+(2326.9329247408422+1213.367853\ 90738162*ln(n)+97.8223892324735979*(ln(n)+.57721566490153286061)^2)*n^5+(2294.2\ 871959552956+185945.0498331263185*ln(n)-126589.87441256318185*(ln(n)+.577215664\ 90153286061)^2+17640.*(ln(n)+.57721566490153286061)^3)*n^2+(23768.1541687017570 +57077.3205038160383*ln(n)-9764.50831093387592*(ln(n)+.57721566490153286061)^2+ 5880.*(ln(n)+.57721566490153286061)^3)*n^3+(-137554.59122817500533+367229.14829\ 73194947*ln(n)-239461.62025709624391*(ln(n)+.57721566490153286061)^2+17640.*(ln (n)+.57721566490153286061)^3)*n+8.648240250323651*n^7+(-236.33207699179703-51.1\ 97316780368167*ln(n))*n^6+243087.22581051344207*ln(n)+5880.*(ln(n)+.57721566490\ 153286061)^3-118766.86066158217937*(ln(n)+.57721566490153286061)^2 The limit of the scaled , 7, -th moment is / 2 4 | 30532750703 1607347 Pi 2660 Pi |- ----------- + ----------- - -------- \ 81000 54 3 2 4 + (1265600/9 - 15680 Pi + 2240/3 Pi ) Zeta(3) 4 2 + 1/90 (63840 - 53760 Zeta(3)) Pi + (-10752 Pi + 112896) Zeta(5) \ / 2\7/2 | / | 2 Pi | + 92160 Zeta(7)| / |7 - -----| / / \ 3 / in Maple format this is: (-30532750703/81000+1607347/54*Pi^2-2660/3*Pi^4+(1265600/9-15680*Pi^2+2240/3*Pi ^4)*Zeta(3)+1/90*(63840-53760*Zeta(3))*Pi^4+(-10752*Pi^2+112896)*Zeta(5)+92160* Zeta(7))/(7-2/3*Pi^2)^(7/2) in LaTex this is: {\frac {-{\frac {30532750703}{81000}}+{\frac {1607347}{54}}\,{\pi }^{2} -{\frac {2660}{3}}\,{\pi }^{4}+ \left( {\frac {1265600}{9}}-15680\,{ \pi }^{2}+{\frac {2240}{3}}\,{\pi }^{4} \right) \zeta \left( 3 \right) +{\frac {1}{90}}\, \left( 63840-53760\,\zeta \left( 3 \right) \right) {\pi }^{4}+ \left( -10752\,{\pi }^{2}+112896 \right) \zeta \left( 5 \right) +92160\,\zeta \left( 7 \right) }{ \left( 7-2/3 \,{\pi }^{2} \right) ^{7/2}}} In floating point, this is , 7, -th moment is 179.72191973561786840 in Maple format this is: 179.72191973561786840 in LaTex this is: 179.72191973561786840 Theorem No. , 8, : The , 8, -th moment about the mean of the random variabl\ e "number of comparisons in Quicksort applied to lists of length n" is 7 6 5 n (90558126238639 n + 874593379118567 n + 3765950297083317 n 4 3 2 + 9448880537396279 n + 15222514193079466 n + 16203885880146258 n + 11083988704917328 n + 4441528864215146)/14883750 - 2/675 (n + 1) ( 6 5 4 3 519753619 n + 4227400499 n + 15454641825 n + 31080886165 n 2 + 38860221806 n + 28165993086 n + 10271667375) H[1](n) + 28/3 4 3 2 2 2 (22600 n + 128500 n + 429230 n + 544910 n + 559371) (n + 1) H[1](n) 2 3 3 4 4 - 560 (42 n + 78 n + 427) (n + 1) H[1](n) + 1680 (n + 1) H[1](n) + (- 6 5 4 3 4 (519753619 n + 3905931099 n + 12918002825 n + 23472392965 n 2 2 + 25572779106 n + 16029920186 n + 3808624725) (n + 1) /675 + 112/3 4 3 2 3 (22600 n + 119380 n + 337850 n + 397490 n + 305061) (n + 1) H[1](n) 2 4 2 5 3 - 1120 (126 n + 234 n + 919) (n + 1) H[1](n) + 13440 (n + 1) H[1](n) ) / | H[2](n) + | \ 4 3 2 4 112 (22600 n + 110260 n + 259910 n + 275030 n + 151227) (n + 1) ------------------------------------------------------------------- 3 \ 2 5 6 2| - 20160 (14 n + 26 n + 69) (n + 1) H[1](n) + 40320 (n + 1) H[1](n) | / 2 6 7 3 H[2](n) + (-13440 %1 (n + 1) + 53760 (n + 1) H[1](n)) H[2](n) 8 4 + 26880 (n + 1) H[2](n) + (- 224 5 4 3 2 (229621 n + 1625435 n + 4444965 n + 6655195 n + 5310824 n + 1959795) 3 3 2 4 (n + 1) /27 + 896 (380 n + 3230 n + 5070 n + 6277) (n + 1) H[1](n) 5 2 - 331520 (n + 1) H[1](n) + 3 2 5 6 (8960 (76 n + 534 n + 806 n + 695) (n + 1) - 1039360 (n + 1) H[1](n)) 7 2 H[2](n) - 752640 (n + 1) H[2](n) ) H[3](n) + 6 7 8 (35840 %1 (n + 1) - 143360 (n + 1) H[1](n) - 286720 (n + 1) H[2](n)) / 2 | H[3](n) + | \ 4 3 2 4 224 (22600 n + 110260 n + 259910 n + 275030 n + 151227) (n + 1) - ------------------------------------------------------------------- 3 2 5 6 2 + 40320 (14 n + 26 n + 69) (n + 1) H[1](n) - 80640 (n + 1) H[1](n) 6 7 + (80640 %1 (n + 1) - 322560 (n + 1) H[1](n)) H[2](n) \ 8 2 7 | - 322560 (n + 1) H[2](n) + 1505280 (n + 1) H[3](n)| H[4](n) / 8 2 + 322560 (n + 1) H[4](n) + ( 3 2 5 6 -10752 (76 n + 534 n + 806 n + 695) (n + 1) + 1247232 (n + 1) H[1](n) 7 8 + 1806336 (n + 1) H[2](n) + 688128 (n + 1) H[3](n)) H[5](n) + 6 7 8 (-107520 %1 (n + 1) + 430080 (n + 1) H[1](n) + 860160 (n + 1) H[2](n)) 7 8 H[6](n) - 2580480 (n + 1) H[7](n) - 1290240 (n + 1) H[8](n) 2 %1 := 14 n + 26 n + 43 and in Maple format 1/14883750*n*(90558126238639*n^7+874593379118567*n^6+3765950297083317*n^5+ 9448880537396279*n^4+15222514193079466*n^3+16203885880146258*n^2+ 11083988704917328*n+4441528864215146)-2/675*(n+1)*(519753619*n^6+4227400499*n^5 +15454641825*n^4+31080886165*n^3+38860221806*n^2+28165993086*n+10271667375)*H[1 ](n)+28/3*(22600*n^4+128500*n^3+429230*n^2+544910*n+559371)*(n+1)^2*H[1](n)^2-\ 560*(42*n^2+78*n+427)*(n+1)^3*H[1](n)^3+1680*(n+1)^4*H[1](n)^4+(-4/675*( 519753619*n^6+3905931099*n^5+12918002825*n^4+23472392965*n^3+25572779106*n^2+ 16029920186*n+3808624725)*(n+1)^2+112/3*(22600*n^4+119380*n^3+337850*n^2+397490 *n+305061)*(n+1)^3*H[1](n)-1120*(126*n^2+234*n+919)*(n+1)^4*H[1](n)^2+13440*(n+ 1)^5*H[1](n)^3)*H[2](n)+(112/3*(22600*n^4+110260*n^3+259910*n^2+275030*n+151227 )*(n+1)^4-20160*(14*n^2+26*n+69)*(n+1)^5*H[1](n)+40320*(n+1)^6*H[1](n)^2)*H[2]( n)^2+(-13440*(14*n^2+26*n+43)*(n+1)^6+53760*(n+1)^7*H[1](n))*H[2](n)^3+26880*(n +1)^8*H[2](n)^4+(-224/27*(229621*n^5+1625435*n^4+4444965*n^3+6655195*n^2+ 5310824*n+1959795)*(n+1)^3+896*(380*n^3+3230*n^2+5070*n+6277)*(n+1)^4*H[1](n)-\ 331520*(n+1)^5*H[1](n)^2+(8960*(76*n^3+534*n^2+806*n+695)*(n+1)^5-1039360*(n+1) ^6*H[1](n))*H[2](n)-752640*(n+1)^7*H[2](n)^2)*H[3](n)+(35840*(14*n^2+26*n+43)*( n+1)^6-143360*(n+1)^7*H[1](n)-286720*(n+1)^8*H[2](n))*H[3](n)^2+(-224/3*(22600* n^4+110260*n^3+259910*n^2+275030*n+151227)*(n+1)^4+40320*(14*n^2+26*n+69)*(n+1) ^5*H[1](n)-80640*(n+1)^6*H[1](n)^2+(80640*(14*n^2+26*n+43)*(n+1)^6-322560*(n+1) ^7*H[1](n))*H[2](n)-322560*(n+1)^8*H[2](n)^2+1505280*(n+1)^7*H[3](n))*H[4](n)+ 322560*(n+1)^8*H[4](n)^2+(-10752*(76*n^3+534*n^2+806*n+695)*(n+1)^5+1247232*(n+ 1)^6*H[1](n)+1806336*(n+1)^7*H[2](n)+688128*(n+1)^8*H[3](n))*H[5](n)+(-107520*( 14*n^2+26*n+43)*(n+1)^6+430080*(n+1)^7*H[1](n)+860160*(n+1)^8*H[2](n))*H[6](n)-\ 2580480*(n+1)^7*H[7](n)-1290240*(n+1)^8*H[8](n) and in LaTex {\frac {1}{14883750}}\,n \left( 90558126238639\,{n}^{7}+874593379118567 \,{n}^{6}+3765950297083317\,{n}^{5}+9448880537396279\,{n}^{4}+ 15222514193079466\,{n}^{3}+16203885880146258\,{n}^{2}+11083988704917328 \,n+4441528864215146 \right) -{\frac {2}{675}}\, \left( n+1 \right) \left( 519753619\,{n}^{6}+4227400499\,{n}^{5}+15454641825\,{n}^{4}+ 31080886165\,{n}^{3}+38860221806\,{n}^{2}+28165993086\,n+10271667375 \right) H_{{1}} \left( n \right) +{\frac {28}{3}}\, \left( 22600\,{n}^ {4}+128500\,{n}^{3}+429230\,{n}^{2}+544910\,n+559371 \right) \left( n+ 1 \right) ^{2} \left( H_{{1}} \left( n \right) \right) ^{2}-560\, \left( 42\,{n}^{2}+78\,n+427 \right) \left( n+1 \right) ^{3} \left( H _{{1}} \left( n \right) \right) ^{3}+1680\, \left( n+1 \right) ^{4} \left( H_{{1}} \left( n \right) \right) ^{4}+ \left( -{\frac {4}{675} }\, \left( 519753619\,{n}^{6}+3905931099\,{n}^{5}+12918002825\,{n}^{4}+ 23472392965\,{n}^{3}+25572779106\,{n}^{2}+16029920186\,n+3808624725 \right) \left( n+1 \right) ^{2}+{\frac {112}{3}}\, \left( 22600\,{n}^ {4}+119380\,{n}^{3}+337850\,{n}^{2}+397490\,n+305061 \right) \left( n+ 1 \right) ^{3}H_{{1}} \left( n \right) -1120\, \left( 126\,{n}^{2}+234 \,n+919 \right) \left( n+1 \right) ^{4} \left( H_{{1}} \left( n \right) \right) ^{2}+13440\, \left( n+1 \right) ^{5} \left( H_{{1}} \left( n \right) \right) ^{3} \right) H_{{2}} \left( n \right) + \left( {\frac {112}{3}}\, \left( 22600\,{n}^{4}+110260\,{n}^{3}+259910 \,{n}^{2}+275030\,n+151227 \right) \left( n+1 \right) ^{4}-20160\, \left( 14\,{n}^{2}+26\,n+69 \right) \left( n+1 \right) ^{5}H_{{1}} \left( n \right) +40320\, \left( n+1 \right) ^{6} \left( H_{{1}} \left( n \right) \right) ^{2} \right) \left( H_{{2}} \left( n \right) \right) ^{2}+ \left( -13440\, \left( 14\,{n}^{2}+26\,n+43 \right) \left( n+1 \right) ^{6}+53760\, \left( n+1 \right) ^{7}H_{{1} } \left( n \right) \right) \left( H_{{2}} \left( n \right) \right) ^ {3}+26880\, \left( n+1 \right) ^{8} \left( H_{{2}} \left( n \right) \right) ^{4}+ \left( -{\frac {224}{27}}\, \left( 229621\,{n}^{5}+ 1625435\,{n}^{4}+4444965\,{n}^{3}+6655195\,{n}^{2}+5310824\,n+1959795 \right) \left( n+1 \right) ^{3}+896\, \left( 380\,{n}^{3}+3230\,{n}^{ 2}+5070\,n+6277 \right) \left( n+1 \right) ^{4}H_{{1}} \left( n \right) -331520\, \left( n+1 \right) ^{5} \left( H_{{1}} \left( n \right) \right) ^{2}+ \left( 8960\, \left( 76\,{n}^{3}+534\,{n}^{2}+ 806\,n+695 \right) \left( n+1 \right) ^{5}-1039360\, \left( n+1 \right) ^{6}H_{{1}} \left( n \right) \right) H_{{2}} \left( n \right) -752640\, \left( n+1 \right) ^{7} \left( H_{{2}} \left( n \right) \right) ^{2} \right) H_{{3}} \left( n \right) + \left( 35840 \, \left( 14\,{n}^{2}+26\,n+43 \right) \left( n+1 \right) ^{6}-143360 \, \left( n+1 \right) ^{7}H_{{1}} \left( n \right) -286720\, \left( n+1 \right) ^{8}H_{{2}} \left( n \right) \right) \left( H_{{3}} \left( n \right) \right) ^{2}+ \left( -{\frac {224}{3}}\, \left( 22600\,{n}^{4 }+110260\,{n}^{3}+259910\,{n}^{2}+275030\,n+151227 \right) \left( n+1 \right) ^{4}+40320\, \left( 14\,{n}^{2}+26\,n+69 \right) \left( n+1 \right) ^{5}H_{{1}} \left( n \right) -80640\, \left( n+1 \right) ^{6} \left( H_{{1}} \left( n \right) \right) ^{2}+ \left( 80640\, \left( 14\,{n}^{2}+26\,n+43 \right) \left( n+1 \right) ^{6}-322560\, \left( n +1 \right) ^{7}H_{{1}} \left( n \right) \right) H_{{2}} \left( n \right) -322560\, \left( n+1 \right) ^{8} \left( H_{{2}} \left( n \right) \right) ^{2}+1505280\, \left( n+1 \right) ^{7}H_{{3}} \left( n \right) \right) H_{{4}} \left( n \right) +322560\, \left( n+1 \right) ^{8} \left( H_{{4}} \left( n \right) \right) ^{2}+ \left( - 10752\, \left( 76\,{n}^{3}+534\,{n}^{2}+806\,n+695 \right) \left( n+1 \right) ^{5}+1247232\, \left( n+1 \right) ^{6}H_{{1}} \left( n \right) +1806336\, \left( n+1 \right) ^{7}H_{{2}} \left( n \right) + 688128\, \left( n+1 \right) ^{8}H_{{3}} \left( n \right) \right) H_{{5 }} \left( n \right) + \left( -107520\, \left( 14\,{n}^{2}+26\,n+43 \right) \left( n+1 \right) ^{6}+430080\, \left( n+1 \right) ^{7}H_{{1 }} \left( n \right) +860160\, \left( n+1 \right) ^{8}H_{{2}} \left( n \right) \right) H_{{6}} \left( n \right) -2580480\, \left( n+1 \right) ^{7}H_{{7}} \left( n \right) -1290240\, \left( n+1 \right) ^{8 }H_{{8}} \left( n \right) This is asymptotically / 2 4 6 8 |90558126238639 1039507238 Pi 632800 Pi 7840 Pi 10256 Pi |-------------- - -------------- + ---------- - -------- - --------- \ 14883750 2025 27 9 135 / 2\ / 2\ | 51435104 340480 Pi | | 143360 Pi | 2 + |- -------- + ----------| Zeta(3) + |501760 - ----------| Zeta(3) \ 27 3 / \ 3 / 2 4 4 (- 5062400/3 + 188160 Pi - 8960 Pi ) Pi + ----------------------------------------- 90 2 6\ / (143360 Pi - 1505280) Pi | 8 | + (-817152 + 688128 Zeta(3)) Zeta(5) + --------------------------| n + | 945 / \ 874593379118567 1039507238 1039507238 gamma --------------- - ---------- ln(n) - ---------------- 14883750 675 675 / 19781753348 2531200 gamma\ 2 + 1/6 |- ----------- + 2531200/3 ln(n) + -------------| Pi \ 675 3 / 4 + 1/36 (22473920/3 - 282240 ln(n) - 282240 gamma) Pi 8 6 82048 Pi + 1/216 (-1478400 + 53760 ln(n) + 53760 gamma) Pi - --------- + 135 / 2 4\ | 518402752 4094720 Pi 62720 Pi | |- --------- + 340480 ln(n) + 340480 gamma + ----------- - ---------| \ 27 3 3 / / 2\ | 1146880 Pi | 2 Zeta(3) + |3942400 - 143360 ln(n) - 143360 gamma - -----------| Zeta(3) + \ 3 / 1/90 (- 44947840/3 + 564480 ln(n) + 564480 gamma 2 4 + 1/6 (8870400 - 322560 ln(n) - 322560 gamma) Pi - 71680 Pi 4 2 + 1505280 Zeta(3)) Pi + (-9827328 + 301056 Pi + 5505024 Zeta(3)) Zeta(5) 2 6 + 1/945 (1146880 Pi + 430080 ln(n) + 430080 gamma - 11827200) Pi \ / | 7 |179330966527777 3164769412 - 2580480 Zeta(7)| n + |--------------- - ---------- ln(n) / \ 708750 225 3164769412 gamma 2 / - ---------------- + 632800/3 (ln(n) + gamma) + 1/6 | 225 \ 9444274952 20964160 gamma 2\ - ---------- + 20964160/3 ln(n) + -------------- - 141120 (ln(n) + gamma) | 75 3 / 2 Pi + 1/36 2 4 (40320 (ln(n) + gamma) - 1935360 ln(n) - 1935360 gamma + 31231200) Pi 8 / 6 287168 Pi | + 1/216 (-5496960 + 376320 ln(n) + 376320 gamma) Pi - ---------- + | 135 \ - 249141088/3 + 4256000 ln(n) + 4256000 gamma 4\ 2 439040 Pi | + 1/6 (37954560 - 1039360 ln(n) - 1039360 gamma) Pi - ----------| Zeta(3) 3 / / 2\ | 4014080 Pi | 2 + |14658560 - 1003520 ln(n) - 1003520 gamma - -----------| Zeta(3) + 1/90 \ 3 / 2 (-62462400 + 3870720 ln(n) + 3870720 gamma - 80640 (ln(n) + gamma) 2 4 + 1/6 (32981760 - 2257920 ln(n) - 2257920 gamma) Pi - 250880 Pi 4 + 10536960 Zeta(3)) Pi + ( 2 -45545472 + 1247232 ln(n) + 1247232 gamma + 2107392 Pi + 19267584 Zeta(3)) Zeta(5) 2 6 + 1/945 (4014080 Pi + 3010560 ln(n) + 3010560 gamma - 43975680) Pi \ / | 6 |1349840076770897 39364084648 - 18063360 Zeta(7)| n + |---------------- - ----------- ln(n) / \ 2126250 675 39364084648 gamma 2 3 - ----------------- + 1621200 (ln(n) + gamma) - 23520 (ln(n) + gamma) + 675 / 212857318856 85544480 gamma 1/6 |- ------------ + 85544480/3 ln(n) + -------------- \ 675 3 2 3\ 2 - 826560 (ln(n) + gamma) + 13440 (ln(n) + gamma) | Pi + 1/36 / 2 4 (231462560/3 - 6834240 ln(n) - 6834240 gamma + 241920 (ln(n) + gamma) ) Pi 8 6 574336 Pi / + 1/216 (-12472320 + 1128960 ln(n) + 1128960 gamma) Pi - ---------- + | 135 \ 5621507584 2 - ---------- + 18161920 ln(n) + 18161920 gamma - 331520 (ln(n) + gamma) 27 2 4\ + 1/6 (96992000 - 6236160 ln(n) - 6236160 gamma) Pi - 439040 Pi | Zeta(3) / / 2\ | 8028160 Pi | 2 + |33259520 - 3010560 ln(n) - 3010560 gamma - -----------| Zeta(3) + 1/90 \ 3 / 2 (- 462925120/3 + 13668480 ln(n) + 13668480 gamma - 483840 (ln(n) + gamma) 2 4 + 1/6 (74833920 - 6773760 ln(n) - 6773760 gamma) Pi - 501760 Pi 4 + 31610880 Zeta(3)) Pi + ( 2 -116390400 + 7483392 ln(n) + 7483392 gamma + 6322176 Pi + 38535168 Zeta(3) ) Zeta(5) 2 6 + 1/945 (8028160 Pi + 9031680 ln(n) + 9031680 gamma - 99778560) Pi \ / | 5 |1087322442362819 18614211196 - 54190080 Zeta(7)| n + |---------------- - ----------- ln(n) / \ 1063125 135 18614211196 gamma 2 - ----------------- + 19847240/3 (ln(n) + gamma) 135 3 4 / 341742271444 - 114240 (ln(n) + gamma) + 1680 (ln(n) + gamma) + 1/6 |- ------------ \ 675 2 + 66893120 ln(n) + 66893120 gamma - 2924320 (ln(n) + gamma) 3\ 2 + 67200 (ln(n) + gamma) | Pi + 1/36 / 2 (366738064/3 - 15019200 ln(n) - 15019200 gamma + 604800 (ln(n) + gamma) ) 8 4 6 143584 Pi Pi + 1/216 (-18480000 + 1881600 ln(n) + 1881600 gamma) Pi - ---------- + 27 / | 9013029536 2 |- ---------- + 42521472 ln(n) + 42521472 gamma - 1657600 (ln(n) + gamma) \ 27 4\ 2 2195200 Pi | + 1/6 (154604800 - 15590400 ln(n) - 15590400 gamma) Pi - -----------| 3 / Zeta(3) / 2\ | 10035200 Pi | 2 + |49280000 - 5017600 ln(n) - 5017600 gamma - ------------| Zeta(3) + \ 3 / 1/90 (- 733476128/3 + 30038400 ln(n) + 30038400 gamma 2 - 1209600 (ln(n) + gamma) 2 4 + 1/6 (110880000 - 11289600 ln(n) - 11289600 gamma) Pi - 627200 Pi 4 + 52684800 Zeta(3)) Pi + (-185525760 + 18708480 ln(n) + 18708480 gamma 2 + 10536960 Pi + 48168960 Zeta(3)) Zeta(5) 2 6 + 1/945 (10035200 Pi + 15052800 ln(n) + 15052800 gamma - 147840000) Pi \ / | 4 |385806806670149 15542468438 - 90316800 Zeta(7)| n + |--------------- - ----------- ln(n) / \ 354375 75 15542468438 gamma 2 - ----------------- + 42892360/3 (ln(n) + gamma) 75 3 4 / 13429314276 - 440720 (ln(n) + gamma) + 6720 (ln(n) + gamma) + 1/6 |- ----------- \ 25 294611632 gamma 2 + 294611632/3 ln(n) + --------------- - 6254080 (ln(n) + gamma) 3 3\ 2 + 134400 (ln(n) + gamma) | Pi + 1/36 / 2 4 (806400 (ln(n) + gamma) - 20563200 ln(n) - 20563200 gamma + 127119552) Pi 8 / 6 574336 Pi | + 1/216 (-17928960 + 1881600 ln(n) + 1881600 gamma) Pi - ---------- + | 135 \ 2 -350956704 + 61669888 ln(n) + 61669888 gamma - 3315200 (ln(n) + gamma) 4\ 2 2195200 Pi | + 1/6 (159093760 - 20787200 ln(n) - 20787200 gamma) Pi - -----------| 3 / / 2\ | 8028160 Pi | 2 Zeta(3) + |47810560 - 5017600 ln(n) - 5017600 gamma - -----------| Zeta(3) \ 3 / + 1/90 (-254239104 + 41126400 ln(n) + 41126400 gamma 2 - 1612800 (ln(n) + gamma) 2 4 + 1/6 (107573760 - 11289600 ln(n) - 11289600 gamma) Pi - 501760 Pi 4 + 52684800 Zeta(3)) Pi + (-190912512 + 24944640 ln(n) + 24944640 gamma 2 + 10536960 Pi + 38535168 Zeta(3)) Zeta(5) 2 6 + 1/945 (8028160 Pi + 15052800 ln(n) + 15052800 gamma - 143431680) Pi \ / | 3 |5541994352458664 134052429784 - 90316800 Zeta(7)| n + |---------------- - ------------ ln(n) / \ 7441875 675 134052429784 gamma 2 3 - ------------------ + 19398596 (ln(n) + gamma) - 871920 (ln(n) + gamma) 675 4 / 245764976812 + 10080 (ln(n) + gamma) + 1/6 |- ------------ + 273896336/3 ln(n) \ 675 273896336 gamma 2 3\ + --------------- - 7365120 (ln(n) + gamma) + 134400 (ln(n) + gamma) | 3 / 2 Pi + 1/36 2 (253947904/3 - 16813440 ln(n) - 16813440 gamma + 604800 (ln(n) + gamma) ) 8 4 6 287168 Pi Pi + 1/216 (-10953600 + 1128960 ln(n) + 1128960 gamma) Pi - ---------- + 135 / 6376619648 2 |- ---------- + 54810112 ln(n) + 54810112 gamma - 3315200 (ln(n) + gamma) \ 27 2 4\ + 1/6 (103165440 - 15590400 ln(n) - 15590400 gamma) Pi - 439040 Pi | / / 2\ | 4014080 Pi | 2 Zeta(3) + |29209600 - 3010560 ln(n) - 3010560 gamma - -----------| Zeta(3) \ 3 / + 1/90 (- 507895808/3 + 33626880 ln(n) + 33626880 gamma 2 - 1209600 (ln(n) + gamma) 2 4 + 1/6 (65721600 - 6773760 ln(n) - 6773760 gamma) Pi - 250880 Pi 4 + 31610880 Zeta(3)) Pi + (-123798528 + 18708480 ln(n) + 18708480 gamma 2 + 6322176 Pi + 19267584 Zeta(3)) Zeta(5) 2 6 + 1/945 (4014080 Pi + 9031680 ln(n) + 9031680 gamma - 87628800) Pi \ / | 2 |2220764432107573 25625106974 - 54190080 Zeta(7)| n + |---------------- - ----------- ln(n) / \ 7441875 225 25625106974 gamma 2 - ----------------- + 46582256/3 (ln(n) + gamma) 225 3 4 / 94588678544 - 761040 (ln(n) + gamma) + 6720 (ln(n) + gamma) + 1/6 |- ----------- \ 675 147019376 gamma 2 + 147019376/3 ln(n) + --------------- - 4379200 (ln(n) + gamma) 3 3\ 2 + 67200 (ln(n) + gamma) | Pi + 1/36 / 2 4 (98553056/3 - 7479360 ln(n) - 7479360 gamma + 241920 (ln(n) + gamma) ) Pi 8 / 6 82048 Pi | + 1/216 (-3816960 + 376320 ln(n) + 376320 gamma) Pi - --------- + | 135 \ 2506606816 2 - ---------- + 27039488 ln(n) + 27039488 gamma - 1657600 (ln(n) + gamma) 27 4\ 2 439040 Pi | + 1/6 (38357760 - 6236160 ln(n) - 6236160 gamma) Pi - ----------| Zeta(3) 3 / / 2\ | 1146880 Pi | 2 + |10178560 - 1003520 ln(n) - 1003520 gamma - -----------| Zeta(3) + 1/90 \ 3 / 2 (- 197106112/3 + 14958720 ln(n) + 14958720 gamma - 483840 (ln(n) + gamma) 2 4 + 1/6 (22901760 - 2257920 ln(n) - 2257920 gamma) Pi - 71680 Pi 4 + 10536960 Zeta(3)) Pi + 2 (-46029312 + 7483392 ln(n) + 7483392 gamma + 2107392 Pi + 5505024 Zeta(3)) Zeta(5) 2 6 + 1/945 (1146880 Pi + 3010560 ln(n) + 3010560 gamma - 30535680) Pi \ | - 18063360 Zeta(7)| n - 30434570 ln(n) - 30434570 gamma / 2 3 + 5220796 (ln(n) + gamma) - 239120 (ln(n) + gamma) 4 3 + 1680 (ln(n) + gamma) + 1/6 (13440 (ln(n) + gamma) 2 - 1029280 (ln(n) + gamma) + 11388944 ln(n) + 11388944 gamma - 22569628) 2 Pi + 2 4 1/36 (40320 (ln(n) + gamma) - 1391040 ln(n) - 1391040 gamma + 5645808) Pi 8 / 6 10256 Pi | + 1/216 (-577920 + 53760 ln(n) + 53760 gamma) Pi - --------- + |-16259040 135 \ 2 + 5624192 ln(n) + 5624192 gamma - 331520 (ln(n) + gamma) 4\ 2 62720 Pi | + 1/6 (6227200 - 1039360 ln(n) - 1039360 gamma) Pi - ---------| Zeta(3) 3 / / 2\ | 143360 Pi | 2 + |1541120 - 143360 ln(n) - 143360 gamma - ----------| Zeta(3) + 1/90 ( \ 3 / 2 -11291616 + 2782080 ln(n) + 2782080 gamma - 80640 (ln(n) + gamma) 2 4 + 1/6 (3467520 - 322560 ln(n) - 322560 gamma) Pi - 8960 Pi 4 + 1505280 Zeta(3)) Pi + 2 (-7472640 + 1247232 ln(n) + 1247232 gamma + 301056 Pi + 688128 Zeta(3)) 2 6 Zeta(5) + 1/945 (143360 Pi + 430080 ln(n) + 430080 gamma - 4623360) Pi - 2580480 Zeta(7) and in Maple format (90558126238639/14883750-1039507238/2025*Pi^2+632800/27*Pi^4-7840/9*Pi^6-10256/ 135*Pi^8+(-51435104/27+340480/3*Pi^2)*Zeta(3)+(501760-143360/3*Pi^2)*Zeta(3)^2+ 1/90*(-5062400/3+188160*Pi^2-8960*Pi^4)*Pi^4+(-817152+688128*Zeta(3))*Zeta(5)+1 /945*(143360*Pi^2-1505280)*Pi^6)*n^8+(874593379118567/14883750-1039507238/675* ln(n)-1039507238/675*gamma+1/6*(-19781753348/675+2531200/3*ln(n)+2531200/3* gamma)*Pi^2+1/36*(22473920/3-282240*ln(n)-282240*gamma)*Pi^4+1/216*(-1478400+ 53760*ln(n)+53760*gamma)*Pi^6-82048/135*Pi^8+(-518402752/27+340480*ln(n)+340480 *gamma+4094720/3*Pi^2-62720/3*Pi^4)*Zeta(3)+(3942400-143360*ln(n)-143360*gamma-\ 1146880/3*Pi^2)*Zeta(3)^2+1/90*(-44947840/3+564480*ln(n)+564480*gamma+1/6*( 8870400-322560*ln(n)-322560*gamma)*Pi^2-71680*Pi^4+1505280*Zeta(3))*Pi^4+(-\ 9827328+301056*Pi^2+5505024*Zeta(3))*Zeta(5)+1/945*(1146880*Pi^2+430080*ln(n)+ 430080*gamma-11827200)*Pi^6-2580480*Zeta(7))*n^7+(179330966527777/708750-\ 3164769412/225*ln(n)-3164769412/225*gamma+632800/3*(ln(n)+gamma)^2+1/6*(-\ 9444274952/75+20964160/3*ln(n)+20964160/3*gamma-141120*(ln(n)+gamma)^2)*Pi^2+1/ 36*(40320*(ln(n)+gamma)^2-1935360*ln(n)-1935360*gamma+31231200)*Pi^4+1/216*(-\ 5496960+376320*ln(n)+376320*gamma)*Pi^6-287168/135*Pi^8+(-249141088/3+4256000* ln(n)+4256000*gamma+1/6*(37954560-1039360*ln(n)-1039360*gamma)*Pi^2-439040/3*Pi ^4)*Zeta(3)+(14658560-1003520*ln(n)-1003520*gamma-4014080/3*Pi^2)*Zeta(3)^2+1/ 90*(-62462400+3870720*ln(n)+3870720*gamma-80640*(ln(n)+gamma)^2+1/6*(32981760-\ 2257920*ln(n)-2257920*gamma)*Pi^2-250880*Pi^4+10536960*Zeta(3))*Pi^4+(-45545472 +1247232*ln(n)+1247232*gamma+2107392*Pi^2+19267584*Zeta(3))*Zeta(5)+1/945*( 4014080*Pi^2+3010560*ln(n)+3010560*gamma-43975680)*Pi^6-18063360*Zeta(7))*n^6+( 1349840076770897/2126250-39364084648/675*ln(n)-39364084648/675*gamma+1621200*( ln(n)+gamma)^2-23520*(ln(n)+gamma)^3+1/6*(-212857318856/675+85544480/3*ln(n)+ 85544480/3*gamma-826560*(ln(n)+gamma)^2+13440*(ln(n)+gamma)^3)*Pi^2+1/36*( 231462560/3-6834240*ln(n)-6834240*gamma+241920*(ln(n)+gamma)^2)*Pi^4+1/216*(-\ 12472320+1128960*ln(n)+1128960*gamma)*Pi^6-574336/135*Pi^8+(-5621507584/27+ 18161920*ln(n)+18161920*gamma-331520*(ln(n)+gamma)^2+1/6*(96992000-6236160*ln(n )-6236160*gamma)*Pi^2-439040*Pi^4)*Zeta(3)+(33259520-3010560*ln(n)-3010560* gamma-8028160/3*Pi^2)*Zeta(3)^2+1/90*(-462925120/3+13668480*ln(n)+13668480* gamma-483840*(ln(n)+gamma)^2+1/6*(74833920-6773760*ln(n)-6773760*gamma)*Pi^2-\ 501760*Pi^4+31610880*Zeta(3))*Pi^4+(-116390400+7483392*ln(n)+7483392*gamma+ 6322176*Pi^2+38535168*Zeta(3))*Zeta(5)+1/945*(8028160*Pi^2+9031680*ln(n)+ 9031680*gamma-99778560)*Pi^6-54190080*Zeta(7))*n^5+(1087322442362819/1063125-\ 18614211196/135*ln(n)-18614211196/135*gamma+19847240/3*(ln(n)+gamma)^2-114240*( ln(n)+gamma)^3+1680*(ln(n)+gamma)^4+1/6*(-341742271444/675+66893120*ln(n)+ 66893120*gamma-2924320*(ln(n)+gamma)^2+67200*(ln(n)+gamma)^3)*Pi^2+1/36*( 366738064/3-15019200*ln(n)-15019200*gamma+604800*(ln(n)+gamma)^2)*Pi^4+1/216*(-\ 18480000+1881600*ln(n)+1881600*gamma)*Pi^6-143584/27*Pi^8+(-9013029536/27+ 42521472*ln(n)+42521472*gamma-1657600*(ln(n)+gamma)^2+1/6*(154604800-15590400* ln(n)-15590400*gamma)*Pi^2-2195200/3*Pi^4)*Zeta(3)+(49280000-5017600*ln(n)-\ 5017600*gamma-10035200/3*Pi^2)*Zeta(3)^2+1/90*(-733476128/3+30038400*ln(n)+ 30038400*gamma-1209600*(ln(n)+gamma)^2+1/6*(110880000-11289600*ln(n)-11289600* gamma)*Pi^2-627200*Pi^4+52684800*Zeta(3))*Pi^4+(-185525760+18708480*ln(n)+ 18708480*gamma+10536960*Pi^2+48168960*Zeta(3))*Zeta(5)+1/945*(10035200*Pi^2+ 15052800*ln(n)+15052800*gamma-147840000)*Pi^6-90316800*Zeta(7))*n^4+( 385806806670149/354375-15542468438/75*ln(n)-15542468438/75*gamma+42892360/3*(ln (n)+gamma)^2-440720*(ln(n)+gamma)^3+6720*(ln(n)+gamma)^4+1/6*(-13429314276/25+ 294611632/3*ln(n)+294611632/3*gamma-6254080*(ln(n)+gamma)^2+134400*(ln(n)+gamma )^3)*Pi^2+1/36*(806400*(ln(n)+gamma)^2-20563200*ln(n)-20563200*gamma+127119552) *Pi^4+1/216*(-17928960+1881600*ln(n)+1881600*gamma)*Pi^6-574336/135*Pi^8+(-\ 350956704+61669888*ln(n)+61669888*gamma-3315200*(ln(n)+gamma)^2+1/6*(159093760-\ 20787200*ln(n)-20787200*gamma)*Pi^2-2195200/3*Pi^4)*Zeta(3)+(47810560-5017600* ln(n)-5017600*gamma-8028160/3*Pi^2)*Zeta(3)^2+1/90*(-254239104+41126400*ln(n)+ 41126400*gamma-1612800*(ln(n)+gamma)^2+1/6*(107573760-11289600*ln(n)-11289600* gamma)*Pi^2-501760*Pi^4+52684800*Zeta(3))*Pi^4+(-190912512+24944640*ln(n)+ 24944640*gamma+10536960*Pi^2+38535168*Zeta(3))*Zeta(5)+1/945*(8028160*Pi^2+ 15052800*ln(n)+15052800*gamma-143431680)*Pi^6-90316800*Zeta(7))*n^3+( 5541994352458664/7441875-134052429784/675*ln(n)-134052429784/675*gamma+19398596 *(ln(n)+gamma)^2-871920*(ln(n)+gamma)^3+10080*(ln(n)+gamma)^4+1/6*(-\ 245764976812/675+273896336/3*ln(n)+273896336/3*gamma-7365120*(ln(n)+gamma)^2+ 134400*(ln(n)+gamma)^3)*Pi^2+1/36*(253947904/3-16813440*ln(n)-16813440*gamma+ 604800*(ln(n)+gamma)^2)*Pi^4+1/216*(-10953600+1128960*ln(n)+1128960*gamma)*Pi^6 -287168/135*Pi^8+(-6376619648/27+54810112*ln(n)+54810112*gamma-3315200*(ln(n)+ gamma)^2+1/6*(103165440-15590400*ln(n)-15590400*gamma)*Pi^2-439040*Pi^4)*Zeta(3 )+(29209600-3010560*ln(n)-3010560*gamma-4014080/3*Pi^2)*Zeta(3)^2+1/90*(-\ 507895808/3+33626880*ln(n)+33626880*gamma-1209600*(ln(n)+gamma)^2+1/6*(65721600 -6773760*ln(n)-6773760*gamma)*Pi^2-250880*Pi^4+31610880*Zeta(3))*Pi^4+(-\ 123798528+18708480*ln(n)+18708480*gamma+6322176*Pi^2+19267584*Zeta(3))*Zeta(5)+ 1/945*(4014080*Pi^2+9031680*ln(n)+9031680*gamma-87628800)*Pi^6-54190080*Zeta(7) )*n^2+(2220764432107573/7441875-25625106974/225*ln(n)-25625106974/225*gamma+ 46582256/3*(ln(n)+gamma)^2-761040*(ln(n)+gamma)^3+6720*(ln(n)+gamma)^4+1/6*(-\ 94588678544/675+147019376/3*ln(n)+147019376/3*gamma-4379200*(ln(n)+gamma)^2+ 67200*(ln(n)+gamma)^3)*Pi^2+1/36*(98553056/3-7479360*ln(n)-7479360*gamma+241920 *(ln(n)+gamma)^2)*Pi^4+1/216*(-3816960+376320*ln(n)+376320*gamma)*Pi^6-82048/ 135*Pi^8+(-2506606816/27+27039488*ln(n)+27039488*gamma-1657600*(ln(n)+gamma)^2+ 1/6*(38357760-6236160*ln(n)-6236160*gamma)*Pi^2-439040/3*Pi^4)*Zeta(3)+( 10178560-1003520*ln(n)-1003520*gamma-1146880/3*Pi^2)*Zeta(3)^2+1/90*(-197106112 /3+14958720*ln(n)+14958720*gamma-483840*(ln(n)+gamma)^2+1/6*(22901760-2257920* ln(n)-2257920*gamma)*Pi^2-71680*Pi^4+10536960*Zeta(3))*Pi^4+(-46029312+7483392* ln(n)+7483392*gamma+2107392*Pi^2+5505024*Zeta(3))*Zeta(5)+1/945*(1146880*Pi^2+ 3010560*ln(n)+3010560*gamma-30535680)*Pi^6-18063360*Zeta(7))*n-30434570*ln(n)-\ 30434570*gamma+5220796*(ln(n)+gamma)^2-239120*(ln(n)+gamma)^3+1680*(ln(n)+gamma )^4+1/6*(13440*(ln(n)+gamma)^3-1029280*(ln(n)+gamma)^2+11388944*ln(n)+11388944* gamma-22569628)*Pi^2+1/36*(40320*(ln(n)+gamma)^2-1391040*ln(n)-1391040*gamma+ 5645808)*Pi^4+1/216*(-577920+53760*ln(n)+53760*gamma)*Pi^6-10256/135*Pi^8+(-\ 16259040+5624192*ln(n)+5624192*gamma-331520*(ln(n)+gamma)^2+1/6*(6227200-\ 1039360*ln(n)-1039360*gamma)*Pi^2-62720/3*Pi^4)*Zeta(3)+(1541120-143360*ln(n)-\ 143360*gamma-143360/3*Pi^2)*Zeta(3)^2+1/90*(-11291616+2782080*ln(n)+2782080* gamma-80640*(ln(n)+gamma)^2+1/6*(3467520-322560*ln(n)-322560*gamma)*Pi^2-8960* Pi^4+1505280*Zeta(3))*Pi^4+(-7472640+1247232*ln(n)+1247232*gamma+301056*Pi^2+ 688128*Zeta(3))*Zeta(5)+1/945*(143360*Pi^2+430080*ln(n)+430080*gamma-4623360)* Pi^6-2580480*Zeta(7) and in LaTex \left( {\frac {90558126238639}{14883750}}-{\frac {1039507238}{2025}}\, {\pi }^{2}+{\frac {632800}{27}}\,{\pi }^{4}-{\frac {7840}{9}}\,{\pi }^{ 6}-{\frac {10256}{135}}\,{\pi }^{8}+ \left( -{\frac {51435104}{27}}+{ \frac {340480}{3}}\,{\pi }^{2} \right) \zeta \left( 3 \right) + \left( 501760-{\frac {143360}{3}}\,{\pi }^{2} \right) \left( \zeta \left( 3 \right) \right) ^{2}+{\frac {1}{90}}\, \left( -{\frac { 5062400}{3}}+188160\,{\pi }^{2}-8960\,{\pi }^{4} \right) {\pi }^{4}+ \left( -817152+688128\,\zeta \left( 3 \right) \right) \zeta \left( 5 \right) +{\frac {1}{945}}\, \left( 143360\,{\pi }^{2}-1505280 \right) {\pi }^{6} \right) {n}^{8}+ \left( {\frac {874593379118567}{ 14883750}}-{\frac {1039507238}{675}}\,\ln \left( n \right) -{\frac { 1039507238}{675}}\,\gamma+1/6\, \left( -{\frac {19781753348}{675}}+{ \frac {2531200}{3}}\,\ln \left( n \right) +{\frac {2531200}{3}}\, \gamma \right) {\pi }^{2}+1/36\, \left( {\frac {22473920}{3}}-282240\, \ln \left( n \right) -282240\,\gamma \right) {\pi }^{4}+{\frac {1}{216 }}\, \left( -1478400+53760\,\ln \left( n \right) +53760\,\gamma \right) {\pi }^{6}-{\frac {82048}{135}}\,{\pi }^{8}+ \left( -{\frac { 518402752}{27}}+340480\,\ln \left( n \right) +340480\,\gamma+{\frac { 4094720}{3}}\,{\pi }^{2}-{\frac {62720}{3}}\,{\pi }^{4} \right) \zeta \left( 3 \right) + \left( 3942400-143360\,\ln \left( n \right) - 143360\,\gamma-{\frac {1146880}{3}}\,{\pi }^{2} \right) \left( \zeta \left( 3 \right) \right) ^{2}+{\frac {1}{90}}\, \left( -{\frac { 44947840}{3}}+564480\,\ln \left( n \right) +564480\,\gamma+1/6\, \left( 8870400-322560\,\ln \left( n \right) -322560\,\gamma \right) { \pi }^{2}-71680\,{\pi }^{4}+1505280\,\zeta \left( 3 \right) \right) { \pi }^{4}+ \left( -9827328+301056\,{\pi }^{2}+5505024\,\zeta \left( 3 \right) \right) \zeta \left( 5 \right) +{\frac {1}{945}}\, \left( 1146880\,{\pi }^{2}+430080\,\ln \left( n \right) +430080\,\gamma- 11827200 \right) {\pi }^{6}-2580480\,\zeta \left( 7 \right) \right) { n}^{7}+ \left( {\frac {179330966527777}{708750}}-{\frac {3164769412}{ 225}}\,\ln \left( n \right) -{\frac {3164769412}{225}}\,\gamma+{\frac {632800}{3}}\, \left( \ln \left( n \right) +\gamma \right) ^{2}+1/6\, \left( -{\frac {9444274952}{75}}+{\frac {20964160}{3}}\,\ln \left( n \right) +{\frac {20964160}{3}}\,\gamma-141120\, \left( \ln \left( n \right) +\gamma \right) ^{2} \right) {\pi }^{2}+1/36\, \left( 40320\, \left( \ln \left( n \right) +\gamma \right) ^{2}-1935360\,\ln \left( n \right) -1935360\,\gamma+31231200 \right) {\pi }^{4}+{\frac { 1}{216}}\, \left( -5496960+376320\,\ln \left( n \right) +376320\, \gamma \right) {\pi }^{6}-{\frac {287168}{135}}\,{\pi }^{8}+ \left( -{ \frac {249141088}{3}}+4256000\,\ln \left( n \right) +4256000\,\gamma+1 /6\, \left( 37954560-1039360\,\ln \left( n \right) -1039360\,\gamma \right) {\pi }^{2}-{\frac {439040}{3}}\,{\pi }^{4} \right) \zeta \left( 3 \right) + \left( 14658560-1003520\,\ln \left( n \right) - 1003520\,\gamma-{\frac {4014080}{3}}\,{\pi }^{2} \right) \left( \zeta \left( 3 \right) \right) ^{2}+{\frac {1}{90}}\, \left( -62462400+ 3870720\,\ln \left( n \right) +3870720\,\gamma-80640\, \left( \ln \left( n \right) +\gamma \right) ^{2}+1/6\, \left( 32981760-2257920\, \ln \left( n \right) -2257920\,\gamma \right) {\pi }^{2}-250880\,{\pi }^{4}+10536960\,\zeta \left( 3 \right) \right) {\pi }^{4}+ \left( - 45545472+1247232\,\ln \left( n \right) +1247232\,\gamma+2107392\,{\pi }^{2}+19267584\,\zeta \left( 3 \right) \right) \zeta \left( 5 \right) +{\frac {1}{945}}\, \left( 4014080\,{\pi }^{2}+3010560\,\ln \left( n \right) +3010560\,\gamma-43975680 \right) {\pi }^{6}-18063360 \,\zeta \left( 7 \right) \right) {n}^{6}+ \left( {\frac { 1349840076770897}{2126250}}-{\frac {39364084648}{675}}\,\ln \left( n \right) -{\frac {39364084648}{675}}\,\gamma+1621200\, \left( \ln \left( n \right) +\gamma \right) ^{2}-23520\, \left( \ln \left( n \right) +\gamma \right) ^{3}+1/6\, \left( -{\frac {212857318856}{675}} +{\frac {85544480}{3}}\,\ln \left( n \right) +{\frac {85544480}{3}}\, \gamma-826560\, \left( \ln \left( n \right) +\gamma \right) ^{2}+13440 \, \left( \ln \left( n \right) +\gamma \right) ^{3} \right) {\pi }^{2} +1/36\, \left( {\frac {231462560}{3}}-6834240\,\ln \left( n \right) - 6834240\,\gamma+241920\, \left( \ln \left( n \right) +\gamma \right) ^ {2} \right) {\pi }^{4}+{\frac {1}{216}}\, \left( -12472320+1128960\, \ln \left( n \right) +1128960\,\gamma \right) {\pi }^{6}-{\frac { 574336}{135}}\,{\pi }^{8}+ \left( -{\frac {5621507584}{27}}+18161920\, \ln \left( n \right) +18161920\,\gamma-331520\, \left( \ln \left( n \right) +\gamma \right) ^{2}+1/6\, \left( 96992000-6236160\,\ln \left( n \right) -6236160\,\gamma \right) {\pi }^{2}-439040\,{\pi }^{4 } \right) \zeta \left( 3 \right) + \left( 33259520-3010560\,\ln \left( n \right) -3010560\,\gamma-{\frac {8028160}{3}}\,{\pi }^{2} \right) \left( \zeta \left( 3 \right) \right) ^{2}+{\frac {1}{90}} \, \left( -{\frac {462925120}{3}}+13668480\,\ln \left( n \right) + 13668480\,\gamma-483840\, \left( \ln \left( n \right) +\gamma \right) ^{2}+1/6\, \left( 74833920-6773760\,\ln \left( n \right) -6773760\, \gamma \right) {\pi }^{2}-501760\,{\pi }^{4}+31610880\,\zeta \left( 3 \right) \right) {\pi }^{4}+ \left( -116390400+7483392\,\ln \left( n \right) +7483392\,\gamma+6322176\,{\pi }^{2}+38535168\,\zeta \left( 3 \right) \right) \zeta \left( 5 \right) +{\frac {1}{945}}\, \left( 8028160\,{\pi }^{2}+9031680\,\ln \left( n \right) +9031680\,\gamma- 99778560 \right) {\pi }^{6}-54190080\,\zeta \left( 7 \right) \right) {n}^{5}+ \left( {\frac {1087322442362819}{1063125}}-{\frac {18614211196 }{135}}\,\ln \left( n \right) -{\frac {18614211196}{135}}\,\gamma+{ \frac {19847240}{3}}\, \left( \ln \left( n \right) +\gamma \right) ^{2 }-114240\, \left( \ln \left( n \right) +\gamma \right) ^{3}+1680\, \left( \ln \left( n \right) +\gamma \right) ^{4}+1/6\, \left( -{ \frac {341742271444}{675}}+66893120\,\ln \left( n \right) +66893120\, \gamma-2924320\, \left( \ln \left( n \right) +\gamma \right) ^{2}+ 67200\, \left( \ln \left( n \right) +\gamma \right) ^{3} \right) {\pi }^{2}+1/36\, \left( {\frac {366738064}{3}}-15019200\,\ln \left( n \right) -15019200\,\gamma+604800\, \left( \ln \left( n \right) + \gamma \right) ^{2} \right) {\pi }^{4}+{\frac {1}{216}}\, \left( - 18480000+1881600\,\ln \left( n \right) +1881600\,\gamma \right) {\pi } ^{6}-{\frac {143584}{27}}\,{\pi }^{8}+ \left( -{\frac {9013029536}{27}} +42521472\,\ln \left( n \right) +42521472\,\gamma-1657600\, \left( \ln \left( n \right) +\gamma \right) ^{2}+1/6\, \left( 154604800- 15590400\,\ln \left( n \right) -15590400\,\gamma \right) {\pi }^{2}-{ \frac {2195200}{3}}\,{\pi }^{4} \right) \zeta \left( 3 \right) + \left( 49280000-5017600\,\ln \left( n \right) -5017600\,\gamma-{ \frac {10035200}{3}}\,{\pi }^{2} \right) \left( \zeta \left( 3 \right) \right) ^{2}+{\frac {1}{90}}\, \left( -{\frac {733476128}{3}} +30038400\,\ln \left( n \right) +30038400\,\gamma-1209600\, \left( \ln \left( n \right) +\gamma \right) ^{2}+1/6\, \left( 110880000- 11289600\,\ln \left( n \right) -11289600\,\gamma \right) {\pi }^{2}- 627200\,{\pi }^{4}+52684800\,\zeta \left( 3 \right) \right) {\pi }^{4 }+ \left( -185525760+18708480\,\ln \left( n \right) +18708480\,\gamma+ 10536960\,{\pi }^{2}+48168960\,\zeta \left( 3 \right) \right) \zeta \left( 5 \right) +{\frac {1}{945}}\, \left( 10035200\,{\pi }^{2}+ 15052800\,\ln \left( n \right) +15052800\,\gamma-147840000 \right) { \pi }^{6}-90316800\,\zeta \left( 7 \right) \right) {n}^{4}+ \left( { \frac {385806806670149}{354375}}-{\frac {15542468438}{75}}\,\ln \left( n \right) -{\frac {15542468438}{75}}\,\gamma+{\frac {42892360}{ 3}}\, \left( \ln \left( n \right) +\gamma \right) ^{2}-440720\, \left( \ln \left( n \right) +\gamma \right) ^{3}+6720\, \left( \ln \left( n \right) +\gamma \right) ^{4}+1/6\, \left( -{\frac { 13429314276}{25}}+{\frac {294611632}{3}}\,\ln \left( n \right) +{ \frac {294611632}{3}}\,\gamma-6254080\, \left( \ln \left( n \right) + \gamma \right) ^{2}+134400\, \left( \ln \left( n \right) +\gamma \right) ^{3} \right) {\pi }^{2}+1/36\, \left( 806400\, \left( \ln \left( n \right) +\gamma \right) ^{2}-20563200\,\ln \left( n \right) -20563200\,\gamma+127119552 \right) {\pi }^{4}+{\frac {1}{216}}\, \left( -17928960+1881600\,\ln \left( n \right) +1881600\,\gamma \right) {\pi }^{6}-{\frac {574336}{135}}\,{\pi }^{8}+ \left( - 350956704+61669888\,\ln \left( n \right) +61669888\,\gamma-3315200\, \left( \ln \left( n \right) +\gamma \right) ^{2}+1/6\, \left( 159093760-20787200\,\ln \left( n \right) -20787200\,\gamma \right) { \pi }^{2}-{\frac {2195200}{3}}\,{\pi }^{4} \right) \zeta \left( 3 \right) + \left( 47810560-5017600\,\ln \left( n \right) -5017600\, \gamma-{\frac {8028160}{3}}\,{\pi }^{2} \right) \left( \zeta \left( 3 \right) \right) ^{2}+{\frac {1}{90}}\, \left( -254239104+41126400\, \ln \left( n \right) +41126400\,\gamma-1612800\, \left( \ln \left( n \right) +\gamma \right) ^{2}+1/6\, \left( 107573760-11289600\,\ln \left( n \right) -11289600\,\gamma \right) {\pi }^{2}-501760\,{\pi }^{ 4}+52684800\,\zeta \left( 3 \right) \right) {\pi }^{4}+ \left( - 190912512+24944640\,\ln \left( n \right) +24944640\,\gamma+10536960\,{ \pi }^{2}+38535168\,\zeta \left( 3 \right) \right) \zeta \left( 5 \right) +{\frac {1}{945}}\, \left( 8028160\,{\pi }^{2}+15052800\,\ln \left( n \right) +15052800\,\gamma-143431680 \right) {\pi }^{6}- 90316800\,\zeta \left( 7 \right) \right) {n}^{3}+ \left( {\frac { 5541994352458664}{7441875}}-{\frac {134052429784}{675}}\,\ln \left( n \right) -{\frac {134052429784}{675}}\,\gamma+19398596\, \left( \ln \left( n \right) +\gamma \right) ^{2}-871920\, \left( \ln \left( n \right) +\gamma \right) ^{3}+10080\, \left( \ln \left( n \right) + \gamma \right) ^{4}+1/6\, \left( -{\frac {245764976812}{675}}+{\frac { 273896336}{3}}\,\ln \left( n \right) +{\frac {273896336}{3}}\,\gamma- 7365120\, \left( \ln \left( n \right) +\gamma \right) ^{2}+134400\, \left( \ln \left( n \right) +\gamma \right) ^{3} \right) {\pi }^{2}+1 /36\, \left( {\frac {253947904}{3}}-16813440\,\ln \left( n \right) - 16813440\,\gamma+604800\, \left( \ln \left( n \right) +\gamma \right) ^{2} \right) {\pi }^{4}+{\frac {1}{216}}\, \left( -10953600+1128960\, \ln \left( n \right) +1128960\,\gamma \right) {\pi }^{6}-{\frac { 287168}{135}}\,{\pi }^{8}+ \left( -{\frac {6376619648}{27}}+54810112\, \ln \left( n \right) +54810112\,\gamma-3315200\, \left( \ln \left( n \right) +\gamma \right) ^{2}+1/6\, \left( 103165440-15590400\,\ln \left( n \right) -15590400\,\gamma \right) {\pi }^{2}-439040\,{\pi }^{ 4} \right) \zeta \left( 3 \right) + \left( 29209600-3010560\,\ln \left( n \right) -3010560\,\gamma-{\frac {4014080}{3}}\,{\pi }^{2} \right) \left( \zeta \left( 3 \right) \right) ^{2}+{\frac {1}{90}} \, \left( -{\frac {507895808}{3}}+33626880\,\ln \left( n \right) + 33626880\,\gamma-1209600\, \left( \ln \left( n \right) +\gamma \right) ^{2}+1/6\, \left( 65721600-6773760\,\ln \left( n \right) - 6773760\,\gamma \right) {\pi }^{2}-250880\,{\pi }^{4}+31610880\,\zeta \left( 3 \right) \right) {\pi }^{4}+ \left( -123798528+18708480\,\ln \left( n \right) +18708480\,\gamma+6322176\,{\pi }^{2}+19267584\, \zeta \left( 3 \right) \right) \zeta \left( 5 \right) +{\frac {1}{ 945}}\, \left( 4014080\,{\pi }^{2}+9031680\,\ln \left( n \right) + 9031680\,\gamma-87628800 \right) {\pi }^{6}-54190080\,\zeta \left( 7 \right) \right) {n}^{2}+ \left( {\frac {2220764432107573}{7441875}}-{ \frac {25625106974}{225}}\,\ln \left( n \right) -{\frac {25625106974}{ 225}}\,\gamma+{\frac {46582256}{3}}\, \left( \ln \left( n \right) + \gamma \right) ^{2}-761040\, \left( \ln \left( n \right) +\gamma \right) ^{3}+6720\, \left( \ln \left( n \right) +\gamma \right) ^{4}+ 1/6\, \left( -{\frac {94588678544}{675}}+{\frac {147019376}{3}}\,\ln \left( n \right) +{\frac {147019376}{3}}\,\gamma-4379200\, \left( \ln \left( n \right) +\gamma \right) ^{2}+67200\, \left( \ln \left( n \right) +\gamma \right) ^{3} \right) {\pi }^{2}+1/36\, \left( {\frac { 98553056}{3}}-7479360\,\ln \left( n \right) -7479360\,\gamma+241920\, \left( \ln \left( n \right) +\gamma \right) ^{2} \right) {\pi }^{4}+{ \frac {1}{216}}\, \left( -3816960+376320\,\ln \left( n \right) +376320 \,\gamma \right) {\pi }^{6}-{\frac {82048}{135}}\,{\pi }^{8}+ \left( -{ \frac {2506606816}{27}}+27039488\,\ln \left( n \right) +27039488\, \gamma-1657600\, \left( \ln \left( n \right) +\gamma \right) ^{2}+1/6 \, \left( 38357760-6236160\,\ln \left( n \right) -6236160\,\gamma \right) {\pi }^{2}-{\frac {439040}{3}}\,{\pi }^{4} \right) \zeta \left( 3 \right) + \left( 10178560-1003520\,\ln \left( n \right) - 1003520\,\gamma-{\frac {1146880}{3}}\,{\pi }^{2} \right) \left( \zeta \left( 3 \right) \right) ^{2}+{\frac {1}{90}}\, \left( -{\frac { 197106112}{3}}+14958720\,\ln \left( n \right) +14958720\,\gamma-483840 \, \left( \ln \left( n \right) +\gamma \right) ^{2}+1/6\, \left( 22901760-2257920\,\ln \left( n \right) -2257920\,\gamma \right) {\pi } ^{2}-71680\,{\pi }^{4}+10536960\,\zeta \left( 3 \right) \right) {\pi }^{4}+ \left( -46029312+7483392\,\ln \left( n \right) +7483392\,\gamma +2107392\,{\pi }^{2}+5505024\,\zeta \left( 3 \right) \right) \zeta \left( 5 \right) +{\frac {1}{945}}\, \left( 1146880\,{\pi }^{2}+ 3010560\,\ln \left( n \right) +3010560\,\gamma-30535680 \right) {\pi } ^{6}-18063360\,\zeta \left( 7 \right) \right) n-30434570\,\ln \left( n \right) -30434570\,\gamma+5220796\, \left( \ln \left( n \right) +\gamma \right) ^{2}-239120\, \left( \ln \left( n \right) + \gamma \right) ^{3}+1680\, \left( \ln \left( n \right) +\gamma \right) ^{4}+1/6\, \left( 13440\, \left( \ln \left( n \right) +\gamma \right) ^{3}-1029280\, \left( \ln \left( n \right) +\gamma \right) ^{ 2}+11388944\,\ln \left( n \right) +11388944\,\gamma-22569628 \right) { \pi }^{2}+1/36\, \left( 40320\, \left( \ln \left( n \right) +\gamma \right) ^{2}-1391040\,\ln \left( n \right) -1391040\,\gamma+5645808 \right) {\pi }^{4}+{\frac {1}{216}}\, \left( -577920+53760\,\ln \left( n \right) +53760\,\gamma \right) {\pi }^{6}-{\frac {10256}{135} }\,{\pi }^{8}+ \left( -16259040+5624192\,\ln \left( n \right) +5624192 \,\gamma-331520\, \left( \ln \left( n \right) +\gamma \right) ^{2}+1/6 \, \left( 6227200-1039360\,\ln \left( n \right) -1039360\,\gamma \right) {\pi }^{2}-{\frac {62720}{3}}\,{\pi }^{4} \right) \zeta \left( 3 \right) + \left( 1541120-143360\,\ln \left( n \right) - 143360\,\gamma-{\frac {143360}{3}}\,{\pi }^{2} \right) \left( \zeta \left( 3 \right) \right) ^{2}+{\frac {1}{90}}\, \left( -11291616+ 2782080\,\ln \left( n \right) +2782080\,\gamma-80640\, \left( \ln \left( n \right) +\gamma \right) ^{2}+1/6\, \left( 3467520-322560\, \ln \left( n \right) -322560\,\gamma \right) {\pi }^{2}-8960\,{\pi }^{ 4}+1505280\,\zeta \left( 3 \right) \right) {\pi }^{4}+ \left( - 7472640+1247232\,\ln \left( n \right) +1247232\,\gamma+301056\,{\pi }^ {2}+688128\,\zeta \left( 3 \right) \right) \zeta \left( 5 \right) +{ \frac {1}{945}}\, \left( 143360\,{\pi }^{2}+430080\,\ln \left( n \right) +430080\,\gamma-4623360 \right) {\pi }^{6}-2580480\,\zeta \left( 7 \right) In floating point this is 8 7 4 -0.51772065663076989851 10 + (0.3315357047354931904 10 + 6720. %1 8 3 - 0.10124749470083260043 10 ln(n) - 650500.43070779918346 %1 7 2 4 + 0.64623116967972692442 10 %1 ) n + (-10904.278378141786 + 10080. %1 7 3 - 0.5253828615764091181 10 ln(n) - 650840.86141559836693 %1 7 2 2 4 + 0.3625694706094351722 10 %1 ) n + (-574183.227324702914 + 6720. %1 7 3 - 0.1589089216707626373 10 ln(n) - 219640.86141559836693 %1 2 3 4 + 461237.767016821285 %1 ) n + (294875.477781008108 + 1680. %1 3 + 48568.221830574521 ln(n) - 3700.43070779918346 %1 2 4 + 140198.0994979658355 %1 ) n + (-72642.750814086508 3 - 46538.311211743637 ln(n) - 1412.086141559836693 %1 2 5 - 6024.78847983946525 %1 ) n + 2 6 (10981.492374466183 + 3825.4937609174271 ln(n) + 619.87421132816455 %1 ) n 4 7 + 1680. %1 + (-839.2687026484899 - 184.6721965032007 ln(n)) n 8 7 + 26.771813151313786 n - 0.65591267700265340105 10 ln(n) 3 7 2 - 217012.08614155983669 %1 + 0.31510119955306053419 10 %1 %1 := ln(n) + 0.57721566490153286061 written in Maple format this is -51772065.663076989851+(3315357.047354931904+6720.*(ln(n)+.57721566490153286061 )^4-10124749.470083260043*ln(n)-650500.43070779918346*(ln(n)+.57721566490153286\ 061)^3+6462311.6967972692442*(ln(n)+.57721566490153286061)^2)*n+(-10904.2783781\ 41786+10080.*(ln(n)+.57721566490153286061)^4-5253828.615764091181*ln(n)-650840.\ 86141559836693*(ln(n)+.57721566490153286061)^3+3625694.706094351722*(ln(n)+.577\ 21566490153286061)^2)*n^2+(-574183.227324702914+6720.*(ln(n)+.57721566490153286\ 061)^4-1589089.216707626373*ln(n)-219640.86141559836693*(ln(n)+.577215664901532\ 86061)^3+461237.767016821285*(ln(n)+.57721566490153286061)^2)*n^3+(294875.47778\ 1008108+1680.*(ln(n)+.57721566490153286061)^4+48568.221830574521*ln(n)-3700.430\ 70779918346*(ln(n)+.57721566490153286061)^3+140198.0994979658355*(ln(n)+.577215\ 66490153286061)^2)*n^4+(-72642.750814086508-46538.311211743637*ln(n)-1412.08614\ 1559836693*(ln(n)+.57721566490153286061)^3-6024.78847983946525*(ln(n)+.57721566\ 490153286061)^2)*n^5+(10981.492374466183+3825.4937609174271*ln(n)+619.874211328\ 16455*(ln(n)+.57721566490153286061)^2)*n^6+1680.*(ln(n)+.57721566490153286061)^ 4+(-839.2687026484899-184.6721965032007*ln(n))*n^7+26.771813151313786*n^8-\ 6559126.7700265340105*ln(n)-217012.08614155983669*(ln(n)+.57721566490153286061) ^3+3151011.9955306053419*(ln(n)+.57721566490153286061)^2 and in LaTex -51772065.663076989851+(3315357.047354931904+6720.*(ln(n)+.57721566490153286061 )^4-10124749.470083260043*ln(n)-650500.43070779918346*(ln(n)+.57721566490153286\ 061)^3+6462311.6967972692442*(ln(n)+.57721566490153286061)^2)*n+(-10904.2783781\ 41786+10080.*(ln(n)+.57721566490153286061)^4-5253828.615764091181*ln(n)-650840.\ 86141559836693*(ln(n)+.57721566490153286061)^3+3625694.706094351722*(ln(n)+.577\ 21566490153286061)^2)*n^2+(-574183.227324702914+6720.*(ln(n)+.57721566490153286\ 061)^4-1589089.216707626373*ln(n)-219640.86141559836693*(ln(n)+.577215664901532\ 86061)^3+461237.767016821285*(ln(n)+.57721566490153286061)^2)*n^3+(294875.47778\ 1008108+1680.*(ln(n)+.57721566490153286061)^4+48568.221830574521*ln(n)-3700.430\ 70779918346*(ln(n)+.57721566490153286061)^3+140198.0994979658355*(ln(n)+.577215\ 66490153286061)^2)*n^4+(-72642.750814086508-46538.311211743637*ln(n)-1412.08614\ 1559836693*(ln(n)+.57721566490153286061)^3-6024.78847983946525*(ln(n)+.57721566\ 490153286061)^2)*n^5+(10981.492374466183+3825.4937609174271*ln(n)+619.874211328\ 16455*(ln(n)+.57721566490153286061)^2)*n^6+1680.*(ln(n)+.57721566490153286061)^ 4+(-839.2687026484899-184.6721965032007*ln(n))*n^7+26.771813151313786*n^8-\ 6559126.7700265340105*ln(n)-217012.08614155983669*(ln(n)+.57721566490153286061) ^3+3151011.9955306053419*(ln(n)+.57721566490153286061)^2 The limit of the scaled , 8, -th moment is / 2 4 6 8 |90558126238639 1039507238 Pi 632800 Pi 7840 Pi 10256 Pi |-------------- - -------------- + ---------- - -------- - --------- \ 14883750 2025 27 9 135 / 2\ / 2\ | 51435104 340480 Pi | | 143360 Pi | 2 + |- -------- + ----------| Zeta(3) + |501760 - ----------| Zeta(3) \ 27 3 / \ 3 / 2 4 4 (- 5062400/3 + 188160 Pi - 8960 Pi ) Pi + ----------------------------------------- 90 2 6\ (143360 Pi - 1505280) Pi | / + (-817152 + 688128 Zeta(3)) Zeta(5) + --------------------------| / 945 / / / 2\4 | 2 Pi | |7 - -----| \ 3 / in Maple format this is: (90558126238639/14883750-1039507238/2025*Pi^2+632800/27*Pi^4-7840/9*Pi^6-10256/ 135*Pi^8+(-51435104/27+340480/3*Pi^2)*Zeta(3)+(501760-143360/3*Pi^2)*Zeta(3)^2+ 1/90*(-5062400/3+188160*Pi^2-8960*Pi^4)*Pi^4+(-817152+688128*Zeta(3))*Zeta(5)+1 /945*(143360*Pi^2-1505280)*Pi^6)/(7-2/3*Pi^2)^4 in LaTex this is: {\frac {{\frac {90558126238639}{14883750}}-{\frac {1039507238}{2025}}\, {\pi }^{2}+{\frac {632800}{27}}\,{\pi }^{4}-{\frac {7840}{9}}\,{\pi }^{ 6}-{\frac {10256}{135}}\,{\pi }^{8}+ \left( -{\frac {51435104}{27}}+{ \frac {340480}{3}}\,{\pi }^{2} \right) \zeta \left( 3 \right) + \left( 501760-{\frac {143360}{3}}\,{\pi }^{2} \right) \left( \zeta \left( 3 \right) \right) ^{2}+{\frac {1}{90}}\, \left( -{\frac { 5062400}{3}}+188160\,{\pi }^{2}-8960\,{\pi }^{4} \right) {\pi }^{4}+ \left( -817152+688128\,\zeta \left( 3 \right) \right) \zeta \left( 5 \right) +{\frac {1}{945}}\, \left( 143360\,{\pi }^{2}-1505280 \right) {\pi }^{6}}{ \left( 7-2/3\,{\pi }^{2} \right) ^{4}}} In floating point, this is , 8, -th moment is 858.20320399000226017 in Maple format this is: 858.20320399000226017 in LaTex this is: 858.20320399000226017 --------------------- This took, 159.275, seconds.