Explicit (Efficient!) Exact Expressions for the Limits of the Scaled moments of the random variable Sum of Parking Functions up to the, 20, -th moment Definition: A (simple) parking function of length n is a permutation of a we\ akly increasing sequence of positive integers such that for all i x[i] <= i For any parking function, p, let X[p] be its sum of entries In this article we will present the limits of the scaled moments from the 3\ -rd through the, 20, -th. 1/2 1/2 1/2 3/2 25 2 Pi 2 Pi ------------- - ---------- 128 16 -------------------------- / Pi \3/2 |- ---- + 5/12| \ 8 / and in Maple format (25/128*2^(1/2)*Pi^(1/2)-1/16*2^(1/2)*Pi^(3/2))/(-1/8*Pi+5/12)^(3/2) and in latex {\frac {{\frac {25}{128}}\,\sqrt {2}\sqrt {\pi }-1/16\,\sqrt {2}{\pi }^ {3/2}}{ \left( -1/8\,\pi +{\frac {5}{12}} \right) ^{3/2}}} 2 221 -3/64 Pi + 5/64 Pi + ---- 1008 -------------------------- / Pi \2 |- ---- + 5/12| \ 8 / and in Maple format (-3/64*Pi^2+5/64*Pi+221/1008)/(-1/8*Pi+5/12)^2 and in latex {\frac {-{\frac {3}{64}}\,{\pi }^{2}+{\frac {5}{64}}\,\pi +{\frac {221} {1008}}}{ \left( -1/8\,\pi +{\frac {5}{12}} \right) ^{2}}} 1/2 1/2 1/2 5/2 1/2 3/2 105845 2 Pi 2 Pi 25 2 Pi ----------------- - ---------- - ------------- 516096 64 1536 ---------------------------------------------- / Pi \5/2 |- ---- + 5/12| \ 8 / and in Maple format (105845/516096*2^(1/2)*Pi^(1/2)-1/64*2^(1/2)*Pi^(5/2)-25/1536*2^(1/2)*Pi^(3/2)) /(-1/8*Pi+5/12)^(5/2) and in latex {\frac {{\frac {105845}{516096}}\,\sqrt {2}\sqrt {\pi }-{\frac {1}{64}} \,\sqrt {2}{\pi }^{5/2}-{\frac {25}{1536}}\,\sqrt {2}{\pi }^{3/2}}{ \left( -1/8\,\pi +{\frac {5}{12}} \right) ^{5/2}}} 82825 35125 3 25 2 ------ + ------ Pi - 5/512 Pi - --- Pi 576576 172032 512 ---------------------------------------- / Pi \3 |- ---- + 5/12| \ 8 / and in Maple format (82825/576576+35125/172032*Pi-5/512*Pi^3-25/512*Pi^2)/(-1/8*Pi+5/12)^3 and in latex {\frac {{\frac {82825}{576576}}+{\frac {35125}{172032}}\,\pi -{\frac {5 }{512}}\,{\pi }^{3}-{\frac {25}{512}}\,{\pi }^{2}}{ \left( -1/8\,\pi +{ \frac {5}{12}} \right) ^{3}}} 1/2 5/2 1/2 3/2 1/2 1/2 1/2 7/2 245 2 Pi 34655 2 Pi 33965825 2 Pi 3 2 Pi - -------------- + ---------------- + ------------------- - ------------ 8192 589824 168689664 1024 ------------------------------------------------------------------------ / Pi \7/2 |- ---- + 5/12| \ 8 / and in Maple format (-245/8192*2^(1/2)*Pi^(5/2)+34655/589824*2^(1/2)*Pi^(3/2)+33965825/168689664*2^ (1/2)*Pi^(1/2)-3/1024*2^(1/2)*Pi^(7/2))/(-1/8*Pi+5/12)^(7/2) and in latex {\frac {-{\frac {245}{8192}}\,\sqrt {2}{\pi }^{5/2}+{\frac {34655}{ 589824}}\,\sqrt {2}{\pi }^{3/2}+{\frac {33965825}{168689664}}\,\sqrt {2 }\sqrt {\pi }-{\frac {3}{1024}}\,\sqrt {2}{\pi }^{7/2}}{ \left( -1/8\, \pi +{\frac {5}{12}} \right) ^{7/2}}} 235 2 256406305 4 175 3 12762625 - ------ Pi + ---------- - 7/4096 Pi - ---- Pi + -------- Pi 147456 2234808576 6144 42172416 --------------------------------------------------------------- / Pi \4 |- ---- + 5/12| \ 8 / and in Maple format (-235/147456*Pi^2+256406305/2234808576-7/4096*Pi^4-175/6144*Pi^3+12762625/ 42172416*Pi)/(-1/8*Pi+5/12)^4 and in latex {\frac {-{\frac {235}{147456}}\,{\pi }^{2}+{\frac {256406305}{ 2234808576}}-{\frac {7}{4096}}\,{\pi }^{4}-{\frac {175}{6144}}\,{\pi }^ {3}+{\frac {12762625}{42172416}}\,\pi }{ \left( -1/8\,\pi +{\frac {5}{ 12}} \right) ^{4}}} / 1/2 9/2 1/2 7/2 1/2 5/2 1/2 3/2 | 2 Pi 195 2 Pi 7307 2 Pi 17084675 2 Pi |- ---------- - -------------- - --------------- + ------------------- \ 2048 16384 262144 112459776 1/2 1/2\ 6978220262525 2 Pi | / / Pi \9/2 + ------------------------| / |- ---- + 5/12| 32546758852608 / / \ 8 / and in Maple format (-1/2048*2^(1/2)*Pi^(9/2)-195/16384*2^(1/2)*Pi^(7/2)-7307/262144*2^(1/2)*Pi^(5/ 2)+17084675/112459776*2^(1/2)*Pi^(3/2)+6978220262525/32546758852608*2^(1/2)*Pi^ (1/2))/(-1/8*Pi+5/12)^(9/2) and in latex {\frac {-{\frac {1}{2048}}\,\sqrt {2}{\pi }^{9/2}-{\frac {195}{16384}} \,\sqrt {2}{\pi }^{7/2}-{\frac {7307}{262144}}\,\sqrt {2}{\pi }^{5/2}+{ \frac {17084675}{112459776}}\,\sqrt {2}{\pi }^{3/2}+{\frac { 6978220262525}{32546758852608}}\,\sqrt {2}\sqrt {\pi }}{ \left( -1/8\, \pi +{\frac {5}{12}} \right) ^{9/2}}} / 75 4 36065 3 10805125 2 304702375 13886296807025 |- ---- Pi - ------ Pi + --------- Pi + ---------- + -------------- Pi \ 8192 786432 112459776 2790982656 32546758852608 5\ / / Pi \5 - 9/32768 Pi | / |- ---- + 5/12| / / \ 8 / and in Maple format (-75/8192*Pi^4-36065/786432*Pi^3+10805125/112459776*Pi^2+304702375/2790982656+ 13886296807025/32546758852608*Pi-9/32768*Pi^5)/(-1/8*Pi+5/12)^5 and in latex {\frac {-{\frac {75}{8192}}\,{\pi }^{4}-{\frac {36065}{786432}}\,{\pi } ^{3}+{\frac {10805125}{112459776}}\,{\pi }^{2}+{\frac {304702375}{ 2790982656}}+{\frac {13886296807025}{32546758852608}}\,\pi -{\frac {9}{ 32768}}\,{\pi }^{5}}{ \left( -1/8\,\pi +{\frac {5}{12}} \right) ^{5}}} / 1/2 9/2 1/2 7/2 1/2 3/2 | 5225 2 Pi 1184975 2 Pi 20654085915475 2 Pi |- --------------- - ------------------ + ------------------------- \ 1572864 44040192 71011110223872 1/2 5/2 1/2 1/2 11/2 1/2\ 203525 2 Pi 882523729466125 2 Pi 5 Pi 2 | / + ----------------- + -------------------------- - -------------| / 163577856 3458658780315648 65536 / / / Pi \11/2 |- ---- + 5/12| \ 8 / and in Maple format (-5225/1572864*2^(1/2)*Pi^(9/2)-1184975/44040192*2^(1/2)*Pi^(7/2)+ 20654085915475/71011110223872*2^(1/2)*Pi^(3/2)+203525/163577856*2^(1/2)*Pi^(5/2 )+882523729466125/3458658780315648*2^(1/2)*Pi^(1/2)-5/65536*Pi^(11/2)*2^(1/2))/ (-1/8*Pi+5/12)^(11/2) and in latex {\frac {-{\frac {5225}{1572864}}\,\sqrt {2}{\pi }^{9/2}-{\frac {1184975 }{44040192}}\,\sqrt {2}{\pi }^{7/2}+{\frac {20654085915475}{ 71011110223872}}\,\sqrt {2}{\pi }^{3/2}+{\frac {203525}{163577856}}\, \sqrt {2}{\pi }^{5/2}+{\frac {882523729466125}{3458658780315648}}\, \sqrt {2}\sqrt {\pi }-{\frac {5}{65536}}\,{\pi }^{11/2}\sqrt {2}}{ \left( -1/8\,\pi +{\frac {5}{12}} \right) ^{11/2}}} / 197065 4 4118525 3 3383894554225 2 11 6 605 5 |- ------- Pi - -------- Pi + -------------- Pi - ------ Pi - ------ Pi \ 7340032 81788928 11835185037312 262144 262144 363330383418125 89585870372335 \ / / Pi \6 + --------------- Pi + ---------------| / |- ---- + 5/12| 576443130052608 739346190815232/ / \ 8 / and in Maple format (-197065/7340032*Pi^4-4118525/81788928*Pi^3+3383894554225/11835185037312*Pi^2-\ 11/262144*Pi^6-605/262144*Pi^5+363330383418125/576443130052608*Pi+ 89585870372335/739346190815232)/(-1/8*Pi+5/12)^6 and in latex {\frac {-{\frac {197065}{7340032}}\,{\pi }^{4}-{\frac {4118525}{ 81788928}}\,{\pi }^{3}+{\frac {3383894554225}{11835185037312}}\,{\pi }^ {2}-{\frac {11}{262144}}\,{\pi }^{6}-{\frac {605}{262144}}\,{\pi }^{5}+ {\frac {363330383418125}{576443130052608}}\,\pi +{\frac {89585870372335 }{739346190815232}}}{ \left( -1/8\,\pi +{\frac {5}{12}} \right) ^{6}}} / 1/2 9/2 1/2 7/2 1/2 5/2 | 25584845 2 Pi 16677625 2 Pi 1283414103665 2 Pi |- ------------------- - ------------------- + ------------------------ \ 2113929216 352321536 14566381584384 1/2 3/2 1/2 13/2 11/2 1/2 570797804206375 2 Pi 3 2 Pi 1625 Pi 2 + -------------------------- - ------------- - ---------------- 1064202701635584 262144 2097152 1/2 1/2\ 324568900626081455515 2 Pi | / / Pi \13/2 + --------------------------------| / |- ---- + 5/12| 954166980160335642624 / / \ 8 / and in Maple format (-25584845/2113929216*2^(1/2)*Pi^(9/2)-16677625/352321536*2^(1/2)*Pi^(7/2)+ 1283414103665/14566381584384*2^(1/2)*Pi^(5/2)+570797804206375/1064202701635584* 2^(1/2)*Pi^(3/2)-3/262144*2^(1/2)*Pi^(13/2)-1625/2097152*Pi^(11/2)*2^(1/2)+ 324568900626081455515/954166980160335642624*2^(1/2)*Pi^(1/2))/(-1/8*Pi+5/12)^( 13/2) and in latex {\frac {-{\frac {25584845}{2113929216}}\,\sqrt {2}{\pi }^{9/2}-{\frac { 16677625}{352321536}}\,\sqrt {2}{\pi }^{7/2}+{\frac {1283414103665}{ 14566381584384}}\,\sqrt {2}{\pi }^{5/2}+{\frac {570797804206375}{ 1064202701635584}}\,\sqrt {2}{\pi }^{3/2}-{\frac {3}{262144}}\,\sqrt {2 }{\pi }^{13/2}-{\frac {1625}{2097152}}\,{\pi }^{11/2}\sqrt {2}+{\frac { 324568900626081455515}{954166980160335642624}}\,\sqrt {2}\sqrt {\pi }}{ \left( -1/8\,\pi +{\frac {5}{12}} \right) ^{13/2}}} / 3139775 4 70143718025 3 3067493 5 13 7 |- -------- Pi - ------------- Pi - --------- Pi - ------- Pi \ 50331648 6242734964736 301989888 2097152 50566333672666475 136693713402998365595 103733710394125 2 + ------------------ + --------------------- Pi + --------------- Pi 328269708721963008 136309568594333663232 152028957376512 3185 6\ / / Pi \7 - ------- Pi | / |- ---- + 5/12| 6291456 / / \ 8 / and in Maple format (-3139775/50331648*Pi^4-70143718025/6242734964736*Pi^3-3067493/301989888*Pi^5-\ 13/2097152*Pi^7+50566333672666475/328269708721963008+136693713402998365595/ 136309568594333663232*Pi+103733710394125/152028957376512*Pi^2-3185/6291456*Pi^6 )/(-1/8*Pi+5/12)^7 and in latex {\frac {-{\frac {3139775}{50331648}}\,{\pi }^{4}-{\frac {70143718025}{ 6242734964736}}\,{\pi }^{3}-{\frac {3067493}{301989888}}\,{\pi }^{5}-{ \frac {13}{2097152}}\,{\pi }^{7}+{\frac {50566333672666475}{ 328269708721963008}}+{\frac {136693713402998365595}{ 136309568594333663232}}\,\pi +{\frac {103733710394125}{152028957376512} }\,{\pi }^{2}-{\frac {3185}{6291456}}\,{\pi }^{6}}{ \left( -1/8\,\pi +{ \frac {5}{12}} \right) ^{7}}} / 1/2 9/2 1/2 7/2 15/2 1/2 | 167793875 2 Pi 7819944878825 2 Pi 7 Pi 2 |- -------------------- - ------------------------ - ------------- \ 4831838208 116531052675072 4194304 1/2 5/2 1/2 3/2 259071878946625 2 Pi 1111029764929560034325 2 Pi + -------------------------- + --------------------------------- 810821106008064 1090476548754669305856 11/2 1/2 1/2 1/2 3251365 Pi 2 14836002272587117225133875 2 Pi - ------------------- + ------------------------------------- 805306368 29373076317255772422537216 1/2 13/2\ 5425 2 Pi | / / Pi \15/2 - ----------------| / |- ---- + 5/12| 33554432 / / \ 8 / and in Maple format (-167793875/4831838208*2^(1/2)*Pi^(9/2)-7819944878825/116531052675072*2^(1/2)* Pi^(7/2)-7/4194304*Pi^(15/2)*2^(1/2)+259071878946625/810821106008064*2^(1/2)*Pi ^(5/2)+1111029764929560034325/1090476548754669305856*2^(1/2)*Pi^(3/2)-3251365/ 805306368*Pi^(11/2)*2^(1/2)+14836002272587117225133875/ 29373076317255772422537216*2^(1/2)*Pi^(1/2)-5425/33554432*2^(1/2)*Pi^(13/2))/(-\ 1/8*Pi+5/12)^(15/2) and in latex {\frac {-{\frac {167793875}{4831838208}}\,\sqrt {2}{\pi }^{9/2}-{\frac {7819944878825}{116531052675072}}\,\sqrt {2}{\pi }^{7/2}-{\frac {7}{ 4194304}}\,{\pi }^{15/2}\sqrt {2}+{\frac {259071878946625}{ 810821106008064}}\,\sqrt {2}{\pi }^{5/2}+{\frac {1111029764929560034325 }{1090476548754669305856}}\,\sqrt {2}{\pi }^{3/2}-{\frac {3251365}{ 805306368}}\,{\pi }^{11/2}\sqrt {2}+{\frac {14836002272587117225133875} {29373076317255772422537216}}\,\sqrt {2}\sqrt {\pi }-{\frac {5425}{ 33554432}}\,\sqrt {2}{\pi }^{13/2}}{ \left( -1/8\,\pi +{\frac {5}{12}} \right) ^{15/2}}} /6352379249043129754797875 20999675 5 15 8 425 7 |------------------------- Pi - --------- Pi - -------- Pi - ------- Pi \3671634539656971552817152 603979776 16777216 4194304 213780598957284103175 2 928525 6 1867306572175 4 + --------------------- Pi - --------- Pi - -------------- Pi 136309568594333663232 301989888 14566381584384 51604458158375 3 301890039495141025 \ / / Pi \8 + --------------- Pi + -------------------| / |- ---- + 5/12| 304057914753024 1366294306063712256/ / \ 8 / and in Maple format (6352379249043129754797875/3671634539656971552817152*Pi-20999675/603979776*Pi^5 -15/16777216*Pi^8-425/4194304*Pi^7+213780598957284103175/136309568594333663232* Pi^2-928525/301989888*Pi^6-1867306572175/14566381584384*Pi^4+51604458158375/ 304057914753024*Pi^3+301890039495141025/1366294306063712256)/(-1/8*Pi+5/12)^8 and in latex {\frac {{\frac {6352379249043129754797875}{3671634539656971552817152}} \,\pi -{\frac {20999675}{603979776}}\,{\pi }^{5}-{\frac {15}{16777216}} \,{\pi }^{8}-{\frac {425}{4194304}}\,{\pi }^{7}+{\frac { 213780598957284103175}{136309568594333663232}}\,{\pi }^{2}-{\frac { 928525}{301989888}}\,{\pi }^{6}-{\frac {1867306572175}{14566381584384}} \,{\pi }^{4}+{\frac {51604458158375}{304057914753024}}\,{\pi }^{3}+{ \frac {301890039495141025}{1366294306063712256}}}{ \left( -1/8\,\pi +{ \frac {5}{12}} \right) ^{8}}} / 1/2 9/2 1/2 1/2 | 22056960275975 2 Pi 223786761634486151078111060075 2 Pi |- ------------------------- + ----------------------------------------- \ 246771640958976 269936710617714297066221469696 1/2 5/2 1/2 3/2 119843005345742558215 2 Pi 10573514723585401794057625 2 Pi + -------------------------------- + ------------------------------------- 128291358677019918336 5183484055986312780447744 1/2 13/2 15/2 1/2 1/2 7/2 5462015 2 Pi 3145 Pi 2 895118787323125 2 Pi - ------------------- - ---------------- - -------------------------- 4831838208 100663296 17027243226169344 11/2 1/2 17/2 1/2\ 857478725 Pi 2 Pi 2 | / / Pi \17/2 - --------------------- - -----------| / |- ---- + 5/12| 53150220288 4194304 / / \ 8 / and in Maple format (-22056960275975/246771640958976*2^(1/2)*Pi^(9/2)+ 223786761634486151078111060075/269936710617714297066221469696*2^(1/2)*Pi^(1/2)+ 119843005345742558215/128291358677019918336*2^(1/2)*Pi^(5/2)+ 10573514723585401794057625/5183484055986312780447744*2^(1/2)*Pi^(3/2)-5462015/ 4831838208*2^(1/2)*Pi^(13/2)-3145/100663296*Pi^(15/2)*2^(1/2)-895118787323125/ 17027243226169344*2^(1/2)*Pi^(7/2)-857478725/53150220288*Pi^(11/2)*2^(1/2)-1/ 4194304*Pi^(17/2)*2^(1/2))/(-1/8*Pi+5/12)^(17/2) and in latex {\frac {-{\frac {22056960275975}{246771640958976}}\,\sqrt {2}{\pi }^{9/ 2}+{\frac {223786761634486151078111060075}{ 269936710617714297066221469696}}\,\sqrt {2}\sqrt {\pi }+{\frac { 119843005345742558215}{128291358677019918336}}\,\sqrt {2}{\pi }^{5/2}+{ \frac {10573514723585401794057625}{5183484055986312780447744}}\,\sqrt { 2}{\pi }^{3/2}-{\frac {5462015}{4831838208}}\,\sqrt {2}{\pi }^{13/2}-{ \frac {3145}{100663296}}\,{\pi }^{15/2}\sqrt {2}-{\frac { 895118787323125}{17027243226169344}}\,\sqrt {2}{\pi }^{7/2}-{\frac { 857478725}{53150220288}}\,{\pi }^{11/2}\sqrt {2}-{\frac {1}{4194304}}\, {\pi }^{17/2}\sqrt {2}}{ \left( -1/8\,\pi +{\frac {5}{12}} \right) ^{17 /2}}} / 3033565 7 2917546797455 5 34330765762828916945 3 |- ---------- Pi - -------------- Pi + -------------------- Pi \ 3758096384 27419071217664 42763786225673306112 2110567737271136019629875 2 22597205976715404689219875 + ------------------------- Pi + -------------------------- 575942672887368086716416 63976178898533676647448576 17 9 1275 8 443098644003875 4 - --------- Pi - -------- Pi - ---------------- Pi 134217728 67108864 1891915914018816 13863334946771821231386540925 250242125 6\ / / Pi \9 + ----------------------------- Pi - ----------- Pi | / |- ---- + 5/12| 4284709692344671382003515392 17716740096 / / \ 8 / and in Maple format (-3033565/3758096384*Pi^7-2917546797455/27419071217664*Pi^5+ 34330765762828916945/42763786225673306112*Pi^3+2110567737271136019629875/ 575942672887368086716416*Pi^2+22597205976715404689219875/ 63976178898533676647448576-17/134217728*Pi^9-1275/67108864*Pi^8-443098644003875 /1891915914018816*Pi^4+13863334946771821231386540925/ 4284709692344671382003515392*Pi-250242125/17716740096*Pi^6)/(-1/8*Pi+5/12)^9 and in latex {\frac {-{\frac {3033565}{3758096384}}\,{\pi }^{7}-{\frac { 2917546797455}{27419071217664}}\,{\pi }^{5}+{\frac { 34330765762828916945}{42763786225673306112}}\,{\pi }^{3}+{\frac { 2110567737271136019629875}{575942672887368086716416}}\,{\pi }^{2}+{ \frac {22597205976715404689219875}{63976178898533676647448576}}-{\frac {17}{134217728}}\,{\pi }^{9}-{\frac {1275}{67108864}}\,{\pi }^{8}-{ \frac {443098644003875}{1891915914018816}}\,{\pi }^{4}+{\frac { 13863334946771821231386540925}{4284709692344671382003515392}}\,\pi -{ \frac {250242125}{17716740096}}\,{\pi }^{6}}{ \left( -1/8\,\pi +{\frac {5}{12}} \right) ^{9}}} / 1/2 9/2 1/2 3/2 | 6194459516738875 2 Pi 164386616875124843400712085825 2 Pi |- --------------------------- + ----------------------------------------- \ 28677462275653632 37885854121784462746136346624 1/2 1/2 19/2 1/2 345423011497575748257465277763075125 2 Pi 9 Pi 2 + ----------------------------------------------- - ------------- 231389975656629426151941520641490944 268435456 1/2 5/2 1/2 13/2 1262205434916737272596275 2 Pi 21744707225 2 Pi + ------------------------------------ - ----------------------- 485004356115678388813824 3685081939968 1/2 7/2 11/2 1/2 17480057705573109565 2 Pi 7258795134625 Pi 2 + ------------------------------- - ------------------------- 126040633086195007488 126993593008128 17/2 1/2 15/2 1/2\ 12255 Pi 2 25354531 Pi 2 | / / Pi \19/2 - ----------------- - --------------------| / |- ---- + 5/12| 2147483648 90194313216 / / \ 8 / and in Maple format (-6194459516738875/28677462275653632*2^(1/2)*Pi^(9/2)+ 164386616875124843400712085825/37885854121784462746136346624*2^(1/2)*Pi^(3/2)+ 345423011497575748257465277763075125/231389975656629426151941520641490944*2^(1/ 2)*Pi^(1/2)-9/268435456*Pi^(19/2)*2^(1/2)+1262205434916737272596275/ 485004356115678388813824*2^(1/2)*Pi^(5/2)-21744707225/3685081939968*2^(1/2)*Pi^ (13/2)+17480057705573109565/126040633086195007488*2^(1/2)*Pi^(7/2)-\ 7258795134625/126993593008128*Pi^(11/2)*2^(1/2)-12255/2147483648*Pi^(17/2)*2^(1 /2)-25354531/90194313216*Pi^(15/2)*2^(1/2))/(-1/8*Pi+5/12)^(19/2) and in latex {\frac {-{\frac {6194459516738875}{28677462275653632}}\,\sqrt {2}{\pi } ^{9/2}+{\frac {164386616875124843400712085825}{ 37885854121784462746136346624}}\,\sqrt {2}{\pi }^{3/2}+{\frac { 345423011497575748257465277763075125}{ 231389975656629426151941520641490944}}\,\sqrt {2}\sqrt {\pi }-{\frac {9 }{268435456}}\,{\pi }^{19/2}\sqrt {2}+{\frac {1262205434916737272596275 }{485004356115678388813824}}\,\sqrt {2}{\pi }^{5/2}-{\frac {21744707225 }{3685081939968}}\,\sqrt {2}{\pi }^{13/2}+{\frac {17480057705573109565} {126040633086195007488}}\,\sqrt {2}{\pi }^{7/2}-{\frac {7258795134625}{ 126993593008128}}\,{\pi }^{11/2}\sqrt {2}-{\frac {12255}{2147483648}}\, {\pi }^{17/2}\sqrt {2}-{\frac {25354531}{90194313216}}\,{\pi }^{15/2} \sqrt {2}}{ \left( -1/8\,\pi +{\frac {5}{12}} \right) ^{19/2}}} / 10925 9 8641865 8 61177532375 7 |- ---------- Pi - ----------- Pi - -------------- Pi \ 3221225472 45097156608 12897786789888 4422064940723375 5 21063385071569759225 4 19 10 - ----------------- Pi - -------------------- Pi - ---------- Pi 14338731137826816 63020316543097503744 1073741824 756572450978194734891282320033295625 12120452147224705149184613725 + ------------------------------------ Pi + ----------------------------- 115694987828314713075970760320745472 19459884677136591383719575552 10853120842925 6 298555957148773474817375 3 - --------------- Pi + ------------------------ Pi 190490389512192 103929504881931083317248 168358180619305236952515748375 2\ / / Pi \10 + ------------------------------ Pi | / |- ---- + 5/12| 18942927060892231373068173312 / / \ 8 / and in Maple format (-10925/3221225472*Pi^9-8641865/45097156608*Pi^8-61177532375/12897786789888*Pi^ 7-4422064940723375/14338731137826816*Pi^5-21063385071569759225/ 63020316543097503744*Pi^4-19/1073741824*Pi^10+ 756572450978194734891282320033295625/115694987828314713075970760320745472*Pi+ 12120452147224705149184613725/19459884677136591383719575552-10853120842925/ 190490389512192*Pi^6+298555957148773474817375/103929504881931083317248*Pi^3+ 168358180619305236952515748375/18942927060892231373068173312*Pi^2)/(-1/8*Pi+5/ 12)^10 and in latex {\frac {-{\frac {10925}{3221225472}}\,{\pi }^{9}-{\frac {8641865}{ 45097156608}}\,{\pi }^{8}-{\frac {61177532375}{12897786789888}}\,{\pi } ^{7}-{\frac {4422064940723375}{14338731137826816}}\,{\pi }^{5}-{\frac { 21063385071569759225}{63020316543097503744}}\,{\pi }^{4}-{\frac {19}{ 1073741824}}\,{\pi }^{10}+{\frac {756572450978194734891282320033295625} {115694987828314713075970760320745472}}\,\pi +{\frac { 12120452147224705149184613725}{19459884677136591383719575552}}-{\frac { 10853120842925}{190490389512192}}\,{\pi }^{6}+{\frac { 298555957148773474817375}{103929504881931083317248}}\,{\pi }^{3}+{ \frac {168358180619305236952515748375}{18942927060892231373068173312}} \,{\pi }^{2}}{ \left( -1/8\,\pi +{\frac {5}{12}} \right) ^{10}}}