On the Statistical Distribution of the Total Height on Complete (Ordered) Binary Trees By Shalosh B. Ekhad Let B(n) be the set of ordered (rooted) trees on n LEAVES where each vertex\ may be a leaf or else must have exactly two children Theorem 1: The average is given explicity by / n \ | 4 2 (2 n - 2)! (-1 + 2 n)| |---- - -----------------------| n! (n - 1)! \ 2 n! (n - 1)! / -------------------------------------------- (2 n - 2)! and in Maple notation by (1/2*4^n-2*(2*n-2)!/n!/(n-1)!*(-1+2*n))/(2*n-2)!*n!*(n-1)! 1/2 3 1/2 3 m Pi 2 this is asymptotically, where m=sqrt(n) , 2 m Pi - --------- - 4 m 4 1/2 1/2 1/2 1/2 1/2 1/2 7 Pi 9 Pi 59 Pi 483 Pi 2323 Pi 42801 Pi - ------- + 2 - ------- + -------- + --------- - ---------- - ------------ 64 m 3 5 7 9 11 512 m 16384 m 131072 m 2097152 m 16777216 m 1/2 923923 Pi + -------------- 13 1073741824 m and in Maple notation 2*m^3*Pi^(1/2)-3/4*m*Pi^(1/2)-4*m^2-7/64/m*Pi^(1/2)+2-9/512/m^3*Pi^(1/2)+59/ 16384/m^5*Pi^(1/2)+483/131072/m^7*Pi^(1/2)-2323/2097152/m^9*Pi^(1/2)-42801/ 16777216/m^11*Pi^(1/2)+923923/1073741824/m^13*Pi^(1/2) Theorem 2: The variance is given explicity by / / 2 \ | | n / 9 n\ %1 (-1 + 2 n) (5 n + 7 n - 3)| |%1 |4 |3/2 - ---| + 4/3 ------------------------------| | \ \ 2 / n! (n - 1)! / |-------------------------------------------------------- \ n! (n - 1)! \ | / n \2| | 4 2 %1 (-1 + 2 n)| | 2 2 / 2 - |---- - ---------------| | (n!) ((n - 1)!) / %1 \ 2 n! (n - 1)! / / / %1 := (2 n - 2)! and in Maple notation by ((2*n-2)!/n!/(n-1)!*(4^n*(3/2-9/2*n)+4/3*(2*n-2)!/n!/(n-1)!*(-1+2*n)*(5*n^2+7*n -3))-(1/2*4^n-2*(2*n-2)!/n!/(n-1)!*(-1+2*n))^2)/(2*n-2)!^2*n!^2*(n-1)!^2 6 5 1/2 this is asymptotically, where m=sqrt(n) , (40/3 - 4 Pi) m - 2 m Pi 3 1/2 1/2 4 5 m Pi / Pi \ 2 55 m Pi 3 Pi + (-4 + 3 Pi) m - ---------- + |- ---- - 4/3| m + ---------- - ---- 4 \ 8 / 64 32 1/2 1/2 1/2 65 Pi 27 Pi 229 Pi 27 Pi 955 Pi 171 Pi + -------- - ------ + --------- - ------- - --------- + -------- 512 m 2 3 4 5 6 512 m 16384 m 2048 m 131072 m 16384 m 1/2 1/2 5405 Pi 621 Pi 61385 Pi - ---------- + -------- + ----------- 7 8 9 2097152 m 65536 m 16777216 m and in Maple notation (40/3-4*Pi)*m^6-2*m^5*Pi^(1/2)+(-4+3*Pi)*m^4-5/4*m^3*Pi^(1/2)+(-1/8*Pi-4/3)*m^2 +55/64*m*Pi^(1/2)-3/32*Pi+65/512/m*Pi^(1/2)-27/512/m^2*Pi+229/16384/m^3*Pi^(1/2 )-27/2048/m^4*Pi-955/131072/m^5*Pi^(1/2)+171/16384/m^6*Pi-5405/2097152/m^7*Pi^( 1/2)+621/65536/m^8*Pi+61385/16777216/m^9*Pi^(1/2) Theorem Number, 3, : The , 3, -th moment (about the mean) is given explicity by / / | 2 | n 2 3 |%1 |4 (7/2 - 81/4 n + 141/4 n + 15/2 n ) \ \ 3 2 \ %1 (-1 + 2 n) (63 n + 21 n - 24 n + 5)| / 2 2 - 8/5 ----------------------------------------| / ((n!) ((n - 1)!) ) - n! (n - 1)! / / / n \ | 4 2 %1 (-1 + 2 n)| 3 |---- - ---------------| %1 \ 2 n! (n - 1)! / / 2 \ | n / 9 n\ %1 (-1 + 2 n) (5 n + 7 n - 3)| |4 |3/2 - ---| + 4/3 ------------------------------|/(n! (n - 1)!) \ \ 2 / n! (n - 1)! / / n \3\ | 4 2 %1 (-1 + 2 n)| | 3 3 / 3 + 2 |---- - ---------------| | (n!) ((n - 1)!) / %1 \ 2 n! (n - 1)! / / / %1 := (2 n - 2)! and in Maple notation by ((2*n-2)!^2/n!^2/(n-1)!^2*(4^n*(7/2-81/4*n+141/4*n^2+15/2*n^3)-8/5*(2*n-2)!/n!/ (n-1)!*(-1+2*n)*(63*n^3+21*n^2-24*n+5))-3*(1/2*4^n-2*(2*n-2)!/n!/(n-1)!*(-1+2*n ))*(2*n-2)!/n!/(n-1)!*(4^n*(3/2-9/2*n)+4/3*(2*n-2)!/n!/(n-1)!*(-1+2*n)*(5*n^2+7 *n-3))+2*(1/2*4^n-2*(2*n-2)!/n!/(n-1)!*(-1+2*n))^3)/(2*n-2)!^3*n!^3*(n-1)!^3 (3/2) 1/2 9 this is asymptotically, where m=sqrt(n) , (16 Pi - 50 Pi ) m / 1/2 \ 8 |255 Pi (3/2)| 7 6 + (12 Pi - 208/5) m + |--------- - 18 Pi | m + (3 Pi - 152/5) m \ 4 / / 1/2 3/2\ | 201 Pi 33 Pi | 5 / 69 Pi\ 4 + |- --------- + --------| m + |112/5 - -----| m \ 64 8 / \ 8 / / 1/2 3/2\ | 2123 Pi 45 Pi | 3 / 21 Pi\ 2 + |- ---------- + --------| m + |8/5 + -----| m \ 512 64 / \ 32 / 3/2 1/2 711 Pi 28463 Pi / 3/2 1/2\ - --------- - ----------- |363 Pi 30099 Pi | 225 Pi 16384 131072 + |--------- - -----------| m + ------ + ------------------------- \ 2048 16384 / 512 m 3/2 1/2 27899 Pi 236627 Pi - ----------- + ------------ 405 Pi 262144 2097152 135 Pi + ------- + ---------------------------- + -------- 2 3 4 2048 m m 16384 m 3/2 1/2 82107 Pi 1052333 Pi - ----------- + ------------- 2097152 16777216 3915 Pi + ----------------------------- - -------- 5 6 m 65536 m 3/2 1/2 9206643 Pi 87316347 Pi ------------- - -------------- 134217728 1073741824 13095 Pi + ------------------------------ - ---------- 7 8 m 2097152 m and in Maple notation (16*Pi^(3/2)-50*Pi^(1/2))*m^9+(12*Pi-208/5)*m^8+(255/4*Pi^(1/2)-18*Pi^(3/2))*m^ 7+(3*Pi-152/5)*m^6+(-201/64*Pi^(1/2)+33/8*Pi^(3/2))*m^5+(112/5-69/8*Pi)*m^4+(-\ 2123/512*Pi^(1/2)+45/64*Pi^(3/2))*m^3+(8/5+21/32*Pi)*m^2+(363/2048*Pi^(3/2)-\ 30099/16384*Pi^(1/2))*m+225/512*Pi+(-711/16384*Pi^(3/2)-28463/131072*Pi^(1/2))/ m+405/2048/m^2*Pi+(-27899/262144*Pi^(3/2)+236627/2097152*Pi^(1/2))/m^3+135/ 16384/m^4*Pi+(-82107/2097152*Pi^(3/2)+1052333/16777216*Pi^(1/2))/m^5-3915/65536 /m^6*Pi+(9206643/134217728*Pi^(3/2)-87316347/1073741824*Pi^(1/2))/m^7-13095/ 2097152/m^8*Pi Theorem Number, 4, : The , 4, -th moment (about the mean) is given explicity by / | | / | 3 | n 2 3 4 |%1 |4 (15/2 - 239/4 n + 203 n - 1129/4 n - 337/2 n ) \ \ 5 4 3 2 16 %1 (-1 + 2 n) (2210 n + 24178 n + 904 n - 5503 n + 2151 n - 315) + --- -------------------------------------------------------------------- 315 n! (n - 1)! \ / | / 3 3 2 | | / ((n!) ((n - 1)!) ) - 4 %2 %1 | / / \ n 2 3 4 (7/2 - 81/4 n + 141/4 n + 15/2 n ) 3 2 \ %1 (-1 + 2 n) (63 n + 21 n - 24 n + 5)| / 2 2 - 8/5 ----------------------------------------| / ((n!) ((n - 1)!) ) n! (n - 1)! / / / 2 \ \ 2 | n / 9 n\ %1 (-1 + 2 n) (5 n + 7 n - 3)| | 6 %2 %1 |4 |3/2 - ---| + 4/3 ------------------------------| | \ \ 2 / n! (n - 1)! / 4| + -------------------------------------------------------------- - 3 %2 | n! (n - 1)! / 4 4 / 4 (n!) ((n - 1)!) / %1 / %1 := (2 n - 2)! n 4 2 %1 (-1 + 2 n) %2 := ---- - --------------- 2 n! (n - 1)! and in Maple notation by ((2*n-2)!^3/n!^3/(n-1)!^3*(4^n*(15/2-239/4*n+203*n^2-1129/4*n^3-337/2*n^4)+16/ 315*(2*n-2)!/n!/(n-1)!*(-1+2*n)*(2210*n^5+24178*n^4+904*n^3-5503*n^2+2151*n-315 ))-4*(1/2*4^n-2*(2*n-2)!/n!/(n-1)!*(-1+2*n))*(2*n-2)!^2/n!^2/(n-1)!^2*(4^n*(7/2 -81/4*n+141/4*n^2+15/2*n^3)-8/5*(2*n-2)!/n!/(n-1)!*(-1+2*n)*(63*n^3+21*n^2-24*n +5))+6*(1/2*4^n-2*(2*n-2)!/n!/(n-1)!*(-1+2*n))^2*(2*n-2)!/n!/(n-1)!*(4^n*(3/2-9 /2*n)+4/3*(2*n-2)!/n!/(n-1)!*(-1+2*n)*(5*n^2+7*n-3))-3*(1/2*4^n-2*(2*n-2)!/n!/( n-1)!*(-1+2*n))^4)/(2*n-2)!^4*n!^4*(n-1)!^4 /14144 2 \ 12 this is asymptotically, where m=sqrt(n) , |----- - 48 Pi + 80 Pi| m \ 63 / / 1/2 \ |694 Pi (3/2)| 11 / 2 41824\ 10 + |--------- - 48 Pi | m + |-324 Pi + 72 Pi + -----| m \ 5 / \ 105 / / 1/2 \ | 2757 Pi (3/2)| 9 / 2 37216\ 8 + |- ---------- + 6 Pi | m + |289/2 Pi - 30 Pi + -----| m \ 20 / \ 315 / / 3/2 1/2\ |333 Pi 73797 Pi | 7 /157 Pi 7072\ 6 + |--------- - -----------| m + |------ - ----| m \ 8 320 / \ 8 105 / / 3/2 1/2\ | 927 Pi 244153 Pi | 5 /655 22544 2\ 4 + |- --------- + ------------| m + |--- Pi - ----- + 3/8 Pi | m \ 64 2560 / \128 315 / / 1/2 3/2\ |1178441 Pi 5409 Pi | 3 / 2 208 2575 \ 2 + |------------- - ----------| m + |9/16 Pi + --- - ---- Pi| m \ 81920 2048 / \ 35 512 / / 3/2 1/2\ 2 | 6579 Pi 2164021 Pi | 8979 Pi 27 Pi + |- ---------- + -------------| m - ------- + ------ \ 16384 655360 / 4096 64 3/2 1/2 117825 Pi 1156965 Pi ------------ - ------------- 262144 2097152 1701 Pi + ---------------------------- + -------- m 2 16384 m 1/2 3/2 49408783 Pi 915897 Pi - -------------- + ------------ 83886080 2097152 + ------------------------------- 3 m and in Maple notation (14144/63-48*Pi^2+80*Pi)*m^12+(694/5*Pi^(1/2)-48*Pi^(3/2))*m^11+(-324*Pi+72*Pi^ 2+41824/105)*m^10+(-2757/20*Pi^(1/2)+6*Pi^(3/2))*m^9+(289/2*Pi-30*Pi^2+37216/ 315)*m^8+(333/8*Pi^(3/2)-73797/320*Pi^(1/2))*m^7+(157/8*Pi-7072/105)*m^6+(-927/ 64*Pi^(3/2)+244153/2560*Pi^(1/2))*m^5+(655/128*Pi-22544/315+3/8*Pi^2)*m^4+( 1178441/81920*Pi^(1/2)-5409/2048*Pi^(3/2))*m^3+(9/16*Pi^2+208/35-2575/512*Pi)*m ^2+(-6579/16384*Pi^(3/2)+2164021/655360*Pi^(1/2))*m-8979/4096*Pi+27/64*Pi^2+( 117825/262144*Pi^(3/2)-1156965/2097152*Pi^(1/2))/m+1701/16384/m^2*Pi+(-49408783 /83886080*Pi^(1/2)+915897/2097152*Pi^(3/2))/m^3 Theorem Number, 5, : The , 5, -th moment (about the mean) is given explicity by / | | | 4 / |%1 | \ \ n / 43635 4 6 5 3 9725 2 \ 4 |31/2 + ----- n + 565/4 n + 22817/8 n - 1865 n + ---- n - 1207/8 n| \ 16 16 / 32 - --- %1 (-1 + 2 n) 231 6 5 4 3 2 (25224 n + 110447 n + 184 n - 15350 n + 6203 n - 1968 n + 231)/(n! / \ / 4 4 3 | (n - 1)!)| / ((n!) ((n - 1)!) ) - 5 %2 %1 | / / \ n 2 3 4 4 (15/2 - 239/4 n + 203 n - 1129/4 n - 337/2 n ) 5 4 3 2 16 %1 (-1 + 2 n) (2210 n + 24178 n + 904 n - 5503 n + 2151 n - 315) + --- -------------------------------------------------------------------- 315 n! (n - 1)! \ / | / 3 3 2 2 | | / ((n!) ((n - 1)!) ) + 10 %2 %1 | / / \ n 2 3 4 (7/2 - 81/4 n + 141/4 n + 15/2 n ) 3 2 \ %1 (-1 + 2 n) (63 n + 21 n - 24 n + 5)| / 2 2 - 8/5 ----------------------------------------| / ((n!) ((n - 1)!) ) n! (n - 1)! / / / 2 \ \ 3 | n / 9 n\ %1 (-1 + 2 n) (5 n + 7 n - 3)| | 10 %2 %1 |4 |3/2 - ---| + 4/3 ------------------------------| | \ \ 2 / n! (n - 1)! / 5| - --------------------------------------------------------------- + 4 %2 | n! (n - 1)! / 5 5 / 5 (n!) ((n - 1)!) / %1 / %1 := (2 n - 2)! n 4 2 %1 (-1 + 2 n) %2 := ---- - --------------- 2 n! (n - 1)! and in Maple notation by ((2*n-2)!^4/n!^4/(n-1)!^4*(4^n*(31/2+43635/16*n^4+565/4*n^6+22817/8*n^5-1865*n^ 3+9725/16*n^2-1207/8*n)-32/231*(2*n-2)!/n!/(n-1)!*(-1+2*n)*(25224*n^6+110447*n^ 5+184*n^4-15350*n^3+6203*n^2-1968*n+231))-5*(1/2*4^n-2*(2*n-2)!/n!/(n-1)!*(-1+2 *n))*(2*n-2)!^3/n!^3/(n-1)!^3*(4^n*(15/2-239/4*n+203*n^2-1129/4*n^3-337/2*n^4)+ 16/315*(2*n-2)!/n!/(n-1)!*(-1+2*n)*(2210*n^5+24178*n^4+904*n^3-5503*n^2+2151*n-\ 315))+10*(1/2*4^n-2*(2*n-2)!/n!/(n-1)!*(-1+2*n))^2*(2*n-2)!^2/n!^2/(n-1)!^2*(4^ n*(7/2-81/4*n+141/4*n^2+15/2*n^3)-8/5*(2*n-2)!/n!/(n-1)!*(-1+2*n)*(63*n^3+21*n^ 2-24*n+5))-10*(1/2*4^n-2*(2*n-2)!/n!/(n-1)!*(-1+2*n))^3*(2*n-2)!/n!/(n-1)!*(4^n *(3/2-9/2*n)+4/3*(2*n-2)!/n!/(n-1)!*(-1+2*n)*(5*n^2+7*n-3))+4*(1/2*4^n-2*(2*n-2 )!/n!/(n-1)!*(-1+2*n))^5)/(2*n-2)!^5*n!^5*(n-1)!^5 this is asymptotically, where m=sqrt(n) , / 3/2 1/2\ |400 Pi (5/2) 105845 Pi | 15 |--------- + 128 Pi - ------------| m \ 3 63 / / 1731328 2\ 14 + |276 Pi - ------- + 160 Pi | m \ 693 / / 1/2 \ | (5/2) 14993 Pi (3/2)| 13 + |-240 Pi - ----------- + 1010 Pi | m \ 24 / / 2 613312\ 12 + |1867 Pi - 80 Pi - ------| m \ 99 / / 1/2 3/2 \ |26205547 Pi 25055 Pi (5/2)| 11 + |-------------- - ----------- + 145 Pi | m \ 8064 24 / / 2 1732000\ 10 + |-140 Pi + 873/8 Pi - -------| m \ 693 / / 3/2 5/2 1/2\ |8315 Pi 165 Pi 32238259 Pi | 9 + |---------- - --------- + --------------| m \ 64 8 21504 / / 2 38295 1242176\ 8 + |100 Pi - ----- Pi + -------| m \ 32 693 / / 3/2 5/2 1/2\ |42785 Pi 885 Pi 51201355 Pi | 7 + |----------- - --------- - --------------| m \ 2048 256 98304 / / 2 73471 78656\ 6 + |-5/4 Pi + ----- Pi + -----| m \ 512 99 / / 1/2 5/2 3/2\ | 866193939 Pi 3969 Pi 705735 Pi | 5 + |- --------------- - ---------- + ------------| m \ 1835008 2048 16384 / / 640 2 104965 \ 4 + |- --- - 25/8 Pi + ------ Pi| m \ 693 2048 / / 5/2 1/2 3/2\ | 25835 Pi 610844681 Pi 1567765 Pi | 3 + |- ----------- - --------------- + -------------| m \ 32768 264241152 786432 / / 105 2 7712 458149 \ 2 + |- --- Pi - ---- + ------ Pi| m \ 32 231 16384 / and in Maple notation (400/3*Pi^(3/2)+128*Pi^(5/2)-105845/63*Pi^(1/2))*m^15+(276*Pi-1731328/693+160* Pi^2)*m^14+(-240*Pi^(5/2)-14993/24*Pi^(1/2)+1010*Pi^(3/2))*m^13+(1867*Pi-80*Pi^ 2-613312/99)*m^12+(26205547/8064*Pi^(1/2)-25055/24*Pi^(3/2)+145*Pi^(5/2))*m^11+ (-140*Pi^2+873/8*Pi-1732000/693)*m^10+(8315/64*Pi^(3/2)-165/8*Pi^(5/2)+32238259 /21504*Pi^(1/2))*m^9+(100*Pi^2-38295/32*Pi+1242176/693)*m^8+(42785/2048*Pi^(3/2 )-885/256*Pi^(5/2)-51201355/98304*Pi^(1/2))*m^7+(-5/4*Pi^2+73471/512*Pi+78656/ 99)*m^6+(-866193939/1835008*Pi^(1/2)-3969/2048*Pi^(5/2)+705735/16384*Pi^(3/2))* m^5+(-640/693-25/8*Pi^2+104965/2048*Pi)*m^4+(-25835/32768*Pi^(5/2)-610844681/ 264241152*Pi^(1/2)+1567765/786432*Pi^(3/2))*m^3+(-105/32*Pi^2-7712/231+458149/ 16384*Pi)*m^2 Theorem Number, 6, : The , 6, -th moment (about the mean) is given explicity by / | | | 5 / n / 162831 7 270025 4 6 598257 5 39833 3 |%1 |4 |63/2 - ------ n + ------ n - 378665/8 n - ------ n - ----- n \ \ \ 28 16 16 16 27825 2 30575 \ 64 8 7 + ----- n - ----- n| + ---- %1 (-1 + 2 n) (331300 n + 11114630 n 16 56 / 9009 6 5 4 3 2 + 30810169 n + 3706098 n - 3614193 n - 5018 n - 288419 n + 143589 n \ / 5 5 4 / - 9009)/(n! (n - 1)!)| / ((n!) ((n - 1)!) ) - 6 %2 %1 | / / \ n / 43635 4 6 5 3 9725 2 \ 4 |31/2 + ----- n + 565/4 n + 22817/8 n - 1865 n + ---- n - 1207/8 n| \ 16 16 / 32 - --- %1 (-1 + 2 n) 231 6 5 4 3 2 (25224 n + 110447 n + 184 n - 15350 n + 6203 n - 1968 n + 231)/(n! / \ / 4 4 2 3 | (n - 1)!)| / ((n!) ((n - 1)!) ) + 15 %2 %1 | / / \ n 2 3 4 4 (15/2 - 239/4 n + 203 n - 1129/4 n - 337/2 n ) 5 4 3 2 16 %1 (-1 + 2 n) (2210 n + 24178 n + 904 n - 5503 n + 2151 n - 315) + --- -------------------------------------------------------------------- 315 n! (n - 1)! \ / | / 3 3 3 2 | | / ((n!) ((n - 1)!) ) - 20 %2 %1 | / / \ n 2 3 4 (7/2 - 81/4 n + 141/4 n + 15/2 n ) 3 2 \ %1 (-1 + 2 n) (63 n + 21 n - 24 n + 5)| / 2 2 - 8/5 ----------------------------------------| / ((n!) ((n - 1)!) ) n! (n - 1)! / / / 2 \ \ 4 | n / 9 n\ %1 (-1 + 2 n) (5 n + 7 n - 3)| | 15 %2 %1 |4 |3/2 - ---| + 4/3 ------------------------------| | \ \ 2 / n! (n - 1)! / 6| + --------------------------------------------------------------- - 5 %2 | n! (n - 1)! / 6 6 / 6 (n!) ((n - 1)!) / %1 / %1 := (2 n - 2)! n 4 2 %1 (-1 + 2 n) %2 := ---- - --------------- 2 n! (n - 1)! and in Maple notation by ((2*n-2)!^5/n!^5/(n-1)!^5*(4^n*(63/2-162831/28*n^7+270025/16*n^4-378665/8*n^6-\ 598257/16*n^5-39833/16*n^3+27825/16*n^2-30575/56*n)+64/9009*(2*n-2)!/n!/(n-1)!* (-1+2*n)*(331300*n^8+11114630*n^7+30810169*n^6+3706098*n^5-3614193*n^4-5018*n^3 -288419*n^2+143589*n-9009))-6*(1/2*4^n-2*(2*n-2)!/n!/(n-1)!*(-1+2*n))*(2*n-2)!^ 4/n!^4/(n-1)!^4*(4^n*(31/2+43635/16*n^4+565/4*n^6+22817/8*n^5-1865*n^3+9725/16* n^2-1207/8*n)-32/231*(2*n-2)!/n!/(n-1)!*(-1+2*n)*(25224*n^6+110447*n^5+184*n^4-\ 15350*n^3+6203*n^2-1968*n+231))+15*(1/2*4^n-2*(2*n-2)!/n!/(n-1)!*(-1+2*n))^2*(2 *n-2)!^3/n!^3/(n-1)!^3*(4^n*(15/2-239/4*n+203*n^2-1129/4*n^3-337/2*n^4)+16/315* (2*n-2)!/n!/(n-1)!*(-1+2*n)*(2210*n^5+24178*n^4+904*n^3-5503*n^2+2151*n-315))-\ 20*(1/2*4^n-2*(2*n-2)!/n!/(n-1)!*(-1+2*n))^3*(2*n-2)!^2/n!^2/(n-1)!^2*(4^n*(7/2 -81/4*n+141/4*n^2+15/2*n^3)-8/5*(2*n-2)!/n!/(n-1)!*(-1+2*n)*(63*n^3+21*n^2-24*n +5))+15*(1/2*4^n-2*(2*n-2)!/n!/(n-1)!*(-1+2*n))^4*(2*n-2)!/n!/(n-1)!*(4^n*(3/2-\ 9/2*n)+4/3*(2*n-2)!/n!/(n-1)!*(-1+2*n)*(5*n^2+7*n-3))-5*(1/2*4^n-2*(2*n-2)!/n!/ (n-1)!*(-1+2*n))^6)/(2*n-2)!^6*n!^6*(n-1)!^6 this is asymptotically, where m=sqrt(n) , / 2 140500 42406400 3\ 18 |-1600 Pi + ------ Pi + -------- - 320 Pi | m \ 21 9009 / / 1/2 \ |4684249 Pi (5/2) (3/2)| 17 + |------------- - 480 Pi - 4984 Pi | m \ 231 / /125291008 2 3\ 16 + |--------- - 1920 Pi - 97023/7 Pi + 720 Pi | m \ 3003 / / 1/2 \ |38933753 Pi (5/2) (3/2)| 15 + |-------------- + 420 Pi - 9269 Pi | m \ 1848 / / 2 3 1819733 282063104\ 14 + |4680 Pi - 570 Pi - ------- Pi + ---------| m \ 56 3003 / / 3/2 1/2 5/2\ |116905 Pi 1468879079 Pi 1425 Pi | 13 + |------------ - ---------------- + ----------| m \ 16 29568 4 / / 3 2 972499 509323712\ 12 + |315/2 Pi - 1820 Pi + ------ Pi + ---------| m \ 224 9009 / / 1/2 5/2 3/2\ | 9734541469 Pi 14925 Pi 690145 Pi | 11 + |- ---------------- - ----------- + ------------| m \ 236544 32 128 / / 2 55206005 11607040 105 3\ 10 + |-15/2 Pi + -------- Pi - -------- + --- Pi | m \ 3584 429 32 / / 1/2 5/2 3/2\ |38618915729 Pi 92475 Pi 13304085 Pi | 9 + |----------------- + ----------- - --------------| m \ 2523136 1024 4096 / /315 3 10341823 2 210472000\ 8 + |--- Pi + -------- Pi - 295/2 Pi - ---------| m \128 14336 9009 / / 5/2 3/2 1/2\ |165735 Pi 9861483 Pi 181487590621 Pi | 7 + |------------ - ------------- + ------------------| m \ 8192 32768 20185088 / and in Maple notation (-1600*Pi^2+140500/21*Pi+42406400/9009-320*Pi^3)*m^18+(4684249/231*Pi^(1/2)-480 *Pi^(5/2)-4984*Pi^(3/2))*m^17+(125291008/3003-1920*Pi^2-97023/7*Pi+720*Pi^3)*m^ 16+(38933753/1848*Pi^(1/2)+420*Pi^(5/2)-9269*Pi^(3/2))*m^15+(4680*Pi^2-570*Pi^3 -1819733/56*Pi+282063104/3003)*m^14+(116905/16*Pi^(3/2)-1468879079/29568*Pi^(1/ 2)+1425/4*Pi^(5/2))*m^13+(315/2*Pi^3-1820*Pi^2+972499/224*Pi+509323712/9009)*m^ 12+(-9734541469/236544*Pi^(1/2)-14925/32*Pi^(5/2)+690145/128*Pi^(3/2))*m^11+(-\ 15/2*Pi^2+55206005/3584*Pi-11607040/429+105/32*Pi^3)*m^10+(38618915729/2523136* Pi^(1/2)+92475/1024*Pi^(5/2)-13304085/4096*Pi^(3/2))*m^9+(315/128*Pi^3+10341823 /14336*Pi-295/2*Pi^2-210472000/9009)*m^8+(165735/8192*Pi^(5/2)-9861483/32768*Pi ^(3/2)+181487590621/20185088*Pi^(1/2))*m^7 Theorem Number, 7, : The , 7, -th moment (about the mean) is given explicity by / | | | 6 / n / 9 8 13722379 7 3220021 4 |%1 |4 |127/2 + 19675/6 n + 167623 n + -------- n - ------- n \ \ \ 16 32 23152269 6 4553773 5 1593679 3 2 \ 128 + -------- n - ------- n - ------- n + 45029/2 n - 13835/8 n| - ------ 32 32 96 / 109395 9 8 7 6 %1 (-1 + 2 n) (105889684 n + 1446232944 n + 3247714950 n + 1000233162 n 5 4 3 2 - 448344324 n - 236075874 n + 26229887 n + 34457727 n - 2829816 n \ / 6 6 5 / n / + 109395)/(n! (n - 1)!)| / ((n!) ((n - 1)!) ) - 7 %2 %1 |4 |63/2 / / \ \ 162831 7 270025 4 6 598257 5 39833 3 27825 2 - ------ n + ------ n - 378665/8 n - ------ n - ----- n + ----- n 28 16 16 16 16 30575 \ 64 8 7 6 - ----- n| + ---- %1 (-1 + 2 n) (331300 n + 11114630 n + 30810169 n 56 / 9009 5 4 3 2 + 3706098 n - 3614193 n - 5018 n - 288419 n + 143589 n - 9009)/(n! \ / 5 5 2 4 / (n - 1)!)| / ((n!) ((n - 1)!) ) + 21 %2 %1 | / / \ n / 43635 4 6 5 3 9725 2 \ 4 |31/2 + ----- n + 565/4 n + 22817/8 n - 1865 n + ---- n - 1207/8 n| \ 16 16 / 32 - --- %1 (-1 + 2 n) 231 6 5 4 3 2 (25224 n + 110447 n + 184 n - 15350 n + 6203 n - 1968 n + 231)/(n! / \ / 4 4 3 3 | (n - 1)!)| / ((n!) ((n - 1)!) ) - 35 %2 %1 | / / \ n 2 3 4 4 (15/2 - 239/4 n + 203 n - 1129/4 n - 337/2 n ) 5 4 3 2 16 %1 (-1 + 2 n) (2210 n + 24178 n + 904 n - 5503 n + 2151 n - 315) + --- -------------------------------------------------------------------- 315 n! (n - 1)! \ / | / 3 3 4 2 | | / ((n!) ((n - 1)!) ) + 35 %2 %1 | / / \ n 2 3 4 (7/2 - 81/4 n + 141/4 n + 15/2 n ) 3 2 \ %1 (-1 + 2 n) (63 n + 21 n - 24 n + 5)| / 2 2 - 8/5 ----------------------------------------| / ((n!) ((n - 1)!) ) n! (n - 1)! / / / 2 \ \ 5 | n / 9 n\ %1 (-1 + 2 n) (5 n + 7 n - 3)| | 21 %2 %1 |4 |3/2 - ---| + 4/3 ------------------------------| | \ \ 2 / n! (n - 1)! / 7| - --------------------------------------------------------------- + 6 %2 | n! (n - 1)! / 7 7 / 7 (n!) ((n - 1)!) / %1 / %1 := (2 n - 2)! n 4 2 %1 (-1 + 2 n) %2 := ---- - --------------- 2 n! (n - 1)! and in Maple notation by ((2*n-2)!^6/n!^6/(n-1)!^6*(4^n*(127/2+19675/6*n^9+167623*n^8+13722379/16*n^7-\ 3220021/32*n^4+23152269/32*n^6-4553773/32*n^5-1593679/96*n^3+45029/2*n^2-13835/ 8*n)-128/109395*(2*n-2)!/n!/(n-1)!*(-1+2*n)*(105889684*n^9+1446232944*n^8+ 3247714950*n^7+1000233162*n^6-448344324*n^5-236075874*n^4+26229887*n^3+34457727 *n^2-2829816*n+109395))-7*(1/2*4^n-2*(2*n-2)!/n!/(n-1)!*(-1+2*n))*(2*n-2)!^5/n! ^5/(n-1)!^5*(4^n*(63/2-162831/28*n^7+270025/16*n^4-378665/8*n^6-598257/16*n^5-\ 39833/16*n^3+27825/16*n^2-30575/56*n)+64/9009*(2*n-2)!/n!/(n-1)!*(-1+2*n)*( 331300*n^8+11114630*n^7+30810169*n^6+3706098*n^5-3614193*n^4-5018*n^3-288419*n^ 2+143589*n-9009))+21*(1/2*4^n-2*(2*n-2)!/n!/(n-1)!*(-1+2*n))^2*(2*n-2)!^4/n!^4/ (n-1)!^4*(4^n*(31/2+43635/16*n^4+565/4*n^6+22817/8*n^5-1865*n^3+9725/16*n^2-\ 1207/8*n)-32/231*(2*n-2)!/n!/(n-1)!*(-1+2*n)*(25224*n^6+110447*n^5+184*n^4-\ 15350*n^3+6203*n^2-1968*n+231))-35*(1/2*4^n-2*(2*n-2)!/n!/(n-1)!*(-1+2*n))^3*(2 *n-2)!^3/n!^3/(n-1)!^3*(4^n*(15/2-239/4*n+203*n^2-1129/4*n^3-337/2*n^4)+16/315* (2*n-2)!/n!/(n-1)!*(-1+2*n)*(2210*n^5+24178*n^4+904*n^3-5503*n^2+2151*n-315))+ 35*(1/2*4^n-2*(2*n-2)!/n!/(n-1)!*(-1+2*n))^4*(2*n-2)!^2/n!^2/(n-1)!^2*(4^n*(7/2 -81/4*n+141/4*n^2+15/2*n^3)-8/5*(2*n-2)!/n!/(n-1)!*(-1+2*n)*(63*n^3+21*n^2-24*n +5))-21*(1/2*4^n-2*(2*n-2)!/n!/(n-1)!*(-1+2*n))^5*(2*n-2)!/n!/(n-1)!*(4^n*(3/2-\ 9/2*n)+4/3*(2*n-2)!/n!/(n-1)!*(-1+2*n)*(5*n^2+7*n-3))+6*(1/2*4^n-2*(2*n-2)!/n!/ (n-1)!*(-1+2*n))^7)/(2*n-2)!^7*n!^7*(n-1)!^7 this is asymptotically, where m=sqrt(n) , / 3/2 1/2 \ | (7/2) 138620 Pi 67931650 Pi (5/2)| 21 |768 Pi - ------------ - -------------- + 7840 Pi | m \ 9 1287 / / 2443186 3 2 1153598464\ 20 + |- ------- Pi + 1344 Pi + 31024 Pi - ----------| m \ 33 9945 / / 1/2 3/2\ | 15633449 Pi (7/2) (5/2) 809945 Pi | 19 + |- -------------- - 2016 Pi - 252 Pi + ------------| m \ 44 6 / /11517001 3 2 29749958144\ 18 + |-------- Pi - 1680 Pi + 28560 Pi - -----------| m \ 66 36465 / / 3/2 5/2 1/2\ |44798969 Pi (7/2) 62531 Pi 25865855401 Pi | 17 + |-------------- + 1974 Pi - ----------- - -----------------| m \ 288 4 82368 / / 2 3 10365323 66461267456\ 16 + |-59430 Pi - 630 Pi + -------- Pi - -----------| m \ 16 36465 / / 5/2 7/2 3/2 1/2\ |364595 Pi 3213 Pi 102039541 Pi 506875685785 Pi | 15 + |------------ - ---------- - --------------- + ------------------| m \ 32 4 768 658944 / / 2 33172409 3 137906097152\ 14 + |-5425 Pi + -------- Pi + 3465/2 Pi - ------------| m + \ 704 109395 / / 7/2 5/2 3/2 1/2\ |9681 Pi 1762985 Pi 7531522405 Pi 19977557172581 Pi | 13 |---------- - ------------- - ---------------- + --------------------| m \ 128 1024 73728 21086208 / / 2 10212564608 408146091 21609 3\ 12 + |191149/8 Pi + ----------- - --------- Pi - ----- Pi | m \ 21879 1024 32 / and in Maple notation (768*Pi^(7/2)-138620/9*Pi^(3/2)-67931650/1287*Pi^(1/2)+7840*Pi^(5/2))*m^21+(-\ 2443186/33*Pi+1344*Pi^3+31024*Pi^2-1153598464/9945)*m^20+(-15633449/44*Pi^(1/2) -2016*Pi^(7/2)-252*Pi^(5/2)+809945/6*Pi^(3/2))*m^19+(11517001/66*Pi-1680*Pi^3+ 28560*Pi^2-29749958144/36465)*m^18+(44798969/288*Pi^(3/2)+1974*Pi^(7/2)-62531/4 *Pi^(5/2)-25865855401/82368*Pi^(1/2))*m^17+(-59430*Pi^2-630*Pi^3+10365323/16*Pi -66461267456/36465)*m^16+(364595/32*Pi^(5/2)-3213/4*Pi^(7/2)-102039541/768*Pi^( 3/2)+506875685785/658944*Pi^(1/2))*m^15+(-5425*Pi^2+33172409/704*Pi+3465/2*Pi^3 -137906097152/109395)*m^14+(9681/128*Pi^(7/2)-1762985/1024*Pi^(5/2)-7531522405/ 73728*Pi^(3/2)+19977557172581/21086208*Pi^(1/2))*m^13+(191149/8*Pi^2+ 10212564608/21879-408146091/1024*Pi-21609/32*Pi^3)*m^12 Theorem Number, 8, : The , 8, -th moment (about the mean) is given explicity by / | | | 7 / n / 10 9 284199207 8 557796517 7 |%1 |4 |255/2 - 860131/4 n - 26628503/6 n - --------- n - --------- n \ \ \ 16 32 7386361 4 6 82952807 5 114461335 3 2 + ------- n + 367437 n + -------- n - --------- n + 296505/8 n 16 16 96 \ 256 11 10 + 437985/8 n| + -------- %1 (-1 + 2 n) (10256252200 n + 759125283404 n / 43648605 9 8 7 + 6488988041724 n + 13625670130371 n + 6702304036344 n 6 5 4 3 - 2251820343192 n - 2257445336494 n + 403059409615 n + 402124726704 n 2 \ / 7 - 38541939273 n - 18740302623 n - 43648605)/(n! (n - 1)!)| / ((n!) / / 7 6 / n / 9 8 13722379 7 ((n - 1)!) ) - 8 %2 %1 |4 |127/2 + 19675/6 n + 167623 n + -------- n \ \ 16 3220021 4 23152269 6 4553773 5 1593679 3 2 - ------- n + -------- n - ------- n - ------- n + 45029/2 n 32 32 32 96 \ 128 9 8 - 13835/8 n| - ------ %1 (-1 + 2 n) (105889684 n + 1446232944 n / 109395 7 6 5 4 + 3247714950 n + 1000233162 n - 448344324 n - 236075874 n 3 2 \ / + 26229887 n + 34457727 n - 2829816 n + 109395)/(n! (n - 1)!)| / ( / / 6 6 2 5 / n / 162831 7 270025 4 (n!) ((n - 1)!) ) + 28 %2 %1 |4 |63/2 - ------ n + ------ n \ \ 28 16 6 598257 5 39833 3 27825 2 30575 \ 64 - 378665/8 n - ------ n - ----- n + ----- n - ----- n| + ---- %1 16 16 16 56 / 9009 8 7 6 5 4 (-1 + 2 n) (331300 n + 11114630 n + 30810169 n + 3706098 n - 3614193 n 3 2 \ / 5 - 5018 n - 288419 n + 143589 n - 9009)/(n! (n - 1)!)| / ((n!) / / 5 3 4 / ((n - 1)!) ) - 56 %2 %1 | \ n / 43635 4 6 5 3 9725 2 \ 4 |31/2 + ----- n + 565/4 n + 22817/8 n - 1865 n + ---- n - 1207/8 n| \ 16 16 / 32 - --- %1 (-1 + 2 n) 231 6 5 4 3 2 (25224 n + 110447 n + 184 n - 15350 n + 6203 n - 1968 n + 231)/(n! / \ / 4 4 4 3 | (n - 1)!)| / ((n!) ((n - 1)!) ) + 70 %2 %1 | / / \ n 2 3 4 4 (15/2 - 239/4 n + 203 n - 1129/4 n - 337/2 n ) 5 4 3 2 16 %1 (-1 + 2 n) (2210 n + 24178 n + 904 n - 5503 n + 2151 n - 315) + --- -------------------------------------------------------------------- 315 n! (n - 1)! \ / | / 3 3 5 2 | | / ((n!) ((n - 1)!) ) - 56 %2 %1 | / / \ n 2 3 4 (7/2 - 81/4 n + 141/4 n + 15/2 n ) 3 2 \ %1 (-1 + 2 n) (63 n + 21 n - 24 n + 5)| / 2 2 - 8/5 ----------------------------------------| / ((n!) ((n - 1)!) ) n! (n - 1)! / / / 2 \ \ 6 | n / 9 n\ %1 (-1 + 2 n) (5 n + 7 n - 3)| | 28 %2 %1 |4 |3/2 - ---| + 4/3 ------------------------------| | \ \ 2 / n! (n - 1)! / 8| + --------------------------------------------------------------- - 7 %2 | n! (n - 1)! / 8 8 / 8 (n!) ((n - 1)!) / %1 / %1 := (2 n - 2)! n 4 2 %1 (-1 + 2 n) %2 := ---- - --------------- 2 n! (n - 1)! and in Maple notation by ((2*n-2)!^7/n!^7/(n-1)!^7*(4^n*(255/2-860131/4*n^10-26628503/6*n^9-284199207/16 *n^8-557796517/32*n^7+7386361/16*n^4+367437*n^6+82952807/16*n^5-114461335/96*n^ 3+296505/8*n^2+437985/8*n)+256/43648605*(2*n-2)!/n!/(n-1)!*(-1+2*n)*( 10256252200*n^11+759125283404*n^10+6488988041724*n^9+13625670130371*n^8+ 6702304036344*n^7-2251820343192*n^6-2257445336494*n^5+403059409615*n^4+ 402124726704*n^3-38541939273*n^2-18740302623*n-43648605))-8*(1/2*4^n-2*(2*n-2)! /n!/(n-1)!*(-1+2*n))*(2*n-2)!^6/n!^6/(n-1)!^6*(4^n*(127/2+19675/6*n^9+167623*n^ 8+13722379/16*n^7-3220021/32*n^4+23152269/32*n^6-4553773/32*n^5-1593679/96*n^3+ 45029/2*n^2-13835/8*n)-128/109395*(2*n-2)!/n!/(n-1)!*(-1+2*n)*(105889684*n^9+ 1446232944*n^8+3247714950*n^7+1000233162*n^6-448344324*n^5-236075874*n^4+ 26229887*n^3+34457727*n^2-2829816*n+109395))+28*(1/2*4^n-2*(2*n-2)!/n!/(n-1)!*( -1+2*n))^2*(2*n-2)!^5/n!^5/(n-1)!^5*(4^n*(63/2-162831/28*n^7+270025/16*n^4-\ 378665/8*n^6-598257/16*n^5-39833/16*n^3+27825/16*n^2-30575/56*n)+64/9009*(2*n-2 )!/n!/(n-1)!*(-1+2*n)*(331300*n^8+11114630*n^7+30810169*n^6+3706098*n^5-3614193 *n^4-5018*n^3-288419*n^2+143589*n-9009))-56*(1/2*4^n-2*(2*n-2)!/n!/(n-1)!*(-1+2 *n))^3*(2*n-2)!^4/n!^4/(n-1)!^4*(4^n*(31/2+43635/16*n^4+565/4*n^6+22817/8*n^5-\ 1865*n^3+9725/16*n^2-1207/8*n)-32/231*(2*n-2)!/n!/(n-1)!*(-1+2*n)*(25224*n^6+ 110447*n^5+184*n^4-15350*n^3+6203*n^2-1968*n+231))+70*(1/2*4^n-2*(2*n-2)!/n!/(n -1)!*(-1+2*n))^4*(2*n-2)!^3/n!^3/(n-1)!^3*(4^n*(15/2-239/4*n+203*n^2-1129/4*n^3 -337/2*n^4)+16/315*(2*n-2)!/n!/(n-1)!*(-1+2*n)*(2210*n^5+24178*n^4+904*n^3-5503 *n^2+2151*n-315))-56*(1/2*4^n-2*(2*n-2)!/n!/(n-1)!*(-1+2*n))^5*(2*n-2)!^2/n!^2/ (n-1)!^2*(4^n*(7/2-81/4*n+141/4*n^2+15/2*n^3)-8/5*(2*n-2)!/n!/(n-1)!*(-1+2*n)*( 63*n^3+21*n^2-24*n+5))+28*(1/2*4^n-2*(2*n-2)!/n!/(n-1)!*(-1+2*n))^6*(2*n-2)!/n! /(n-1)!*(4^n*(3/2-9/2*n)+4/3*(2*n-2)!/n!/(n-1)!*(-1+2*n)*(5*n^2+7*n-3))-7*(1/2* 4^n-2*(2*n-2)!/n!/(n-1)!*(-1+2*n))^8)/(2*n-2)!^8*n!^8*(n-1)!^8 this is asymptotically, where m=sqrt(n) , /1050240225280 408404000 3 2 4\ 24 |------------- + --------- Pi - 89600/3 Pi - 15040/9 Pi - 1792 Pi | m \ 8729721 1287 / / 5/2 3/2 1/2 \ | 713664 Pi 3233968 Pi 14077820629 Pi (7/2)| 23 + |- ------------ + ------------- + ----------------- - 3584 Pi | m \ 5 99 9945 / + /43993437294592 2 4 467924216 3\ 22 |-------------- - 2321216/3 Pi + 5376 Pi + --------- Pi + 22400 Pi | m \ 14549535 429 / + / 3/2 5/2 1/2\ | 235246766 Pi (7/2) 231896 Pi 2222928222473 Pi | 21 |- --------------- + 5824 Pi - ------------ + -------------------| m \ 99 5 291720 / / + | \ 2 3 111321594852352 4 5386402547 \ -3937976/9 Pi + 119504/3 Pi + --------------- - 6272 Pi - ---------- Pi| 6235515 1287 / 20 m + / 3/2 5/2 1/2\ | 3036931981 Pi 3018057 Pi (7/2) 19945553570915 Pi | |- ---------------- + ------------- + 196 Pi + --------------------| \ 792 10 2800512 / 19 m + / 2 4 3 67492828633 118748295763456\ |3170180/3 Pi + 3360 Pi - 51212 Pi - ----------- Pi + ---------------| \ 5148 2909907 / 18 m + / 5/2 3/2 7/2 1/2\ | 7250481 Pi 1317109553 Pi 10927 Pi 1762959370996207 Pi | |- ------------- + ---------------- - ----------- - ----------------------| \ 80 576 2 112020480 / 17 m and in Maple notation (1050240225280/8729721+408404000/1287*Pi-89600/3*Pi^3-15040/9*Pi^2-1792*Pi^4)*m ^24+(-713664/5*Pi^(5/2)+3233968/99*Pi^(3/2)+14077820629/9945*Pi^(1/2)-3584*Pi^( 7/2))*m^23+(43993437294592/14549535-2321216/3*Pi^2+5376*Pi^4+467924216/429*Pi+ 22400*Pi^3)*m^22+(-235246766/99*Pi^(3/2)+5824*Pi^(7/2)-231896/5*Pi^(5/2)+ 2222928222473/291720*Pi^(1/2))*m^21+(-3937976/9*Pi^2+119504/3*Pi^3+ 111321594852352/6235515-6272*Pi^4-5386402547/1287*Pi)*m^20+(-3036931981/792*Pi^ (3/2)+3018057/10*Pi^(5/2)+196*Pi^(7/2)+19945553570915/2800512*Pi^(1/2))*m^19+( 3170180/3*Pi^2+3360*Pi^4-51212*Pi^3-67492828633/5148*Pi+118748295763456/2909907 )*m^18+(-7250481/80*Pi^(5/2)+1317109553/576*Pi^(3/2)-10927/2*Pi^(7/2)-\ 1762959370996207/112020480*Pi^(1/2))*m^17 Theorem Number, 9, : The , 9, -th moment (about the mean) is given explicity by / | | | 8 / n / 38234255219 10 54208131033 9 122845422387 8 |%1 |4 |511/2 + ----------- n + ----------- n + ------------ n \ \ \ 320 128 256 16145585 12 7455425931 7 16725648203 4 751249461 11 + -------- n + ---------- n + ----------- n + --------- n 176 160 256 80 127978887571 6 4012381707 5 972410319 3 31713828081 2 - ------------ n - ---------- n + --------- n - ----------- n 640 128 320 3520 3078273 \ 512 12 + ------- n| - --------- %1 (-1 + 2 n) (1048241727120 n 16 / 111546435 11 10 9 + 31226624575240 n + 204201623034104 n + 429879431852561 n 8 7 6 + 276904518461427 n - 77822821450878 n - 122633209301358 n 5 4 3 + 22615990332254 n + 33039275776063 n - 4175143669772 n 2 \ / 8 - 3809693871861 n + 83749453200 n + 111546435)/(n! (n - 1)!)| / ((n!) / / 8 7 / n / 10 9 ((n - 1)!) ) - 9 %2 %1 |4 |255/2 - 860131/4 n - 26628503/6 n \ \ 284199207 8 557796517 7 7386361 4 6 82952807 5 - --------- n - --------- n + ------- n + 367437 n + -------- n 16 32 16 16 114461335 3 2 \ 256 - --------- n + 296505/8 n + 437985/8 n| + -------- %1 (-1 + 2 n) ( 96 / 43648605 11 10 9 8 10256252200 n + 759125283404 n + 6488988041724 n + 13625670130371 n 7 6 5 4 + 6702304036344 n - 2251820343192 n - 2257445336494 n + 403059409615 n 3 2 + 402124726704 n - 38541939273 n - 18740302623 n - 43648605)/(n! \ / 7 7 2 6 / n / 9 (n - 1)!)| / ((n!) ((n - 1)!) ) + 36 %2 %1 |4 |127/2 + 19675/6 n / / \ \ 8 13722379 7 3220021 4 23152269 6 4553773 5 + 167623 n + -------- n - ------- n + -------- n - ------- n 16 32 32 32 1593679 3 2 \ 128 - ------- n + 45029/2 n - 13835/8 n| - ------ %1 (-1 + 2 n) ( 96 / 109395 9 8 7 6 5 105889684 n + 1446232944 n + 3247714950 n + 1000233162 n - 448344324 n 4 3 2 - 236075874 n + 26229887 n + 34457727 n - 2829816 n + 109395)/(n! \ / 6 6 3 5 / n / 162831 7 (n - 1)!)| / ((n!) ((n - 1)!) ) - 84 %2 %1 |4 |63/2 - ------ n / / \ \ 28 270025 4 6 598257 5 39833 3 27825 2 30575 \ + ------ n - 378665/8 n - ------ n - ----- n + ----- n - ----- n| + 16 16 16 16 56 / 64 8 7 6 5 ---- %1 (-1 + 2 n) (331300 n + 11114630 n + 30810169 n + 3706098 n 9009 4 3 2 \ / - 3614193 n - 5018 n - 288419 n + 143589 n - 9009)/(n! (n - 1)!)| / ( / / 5 5 4 4 / (n!) ((n - 1)!) ) + 126 %2 %1 | \ n / 43635 4 6 5 3 9725 2 \ 4 |31/2 + ----- n + 565/4 n + 22817/8 n - 1865 n + ---- n - 1207/8 n| \ 16 16 / 32 - --- %1 (-1 + 2 n) 231 6 5 4 3 2 (25224 n + 110447 n + 184 n - 15350 n + 6203 n - 1968 n + 231)/(n! / \ / 4 4 5 3 | (n - 1)!)| / ((n!) ((n - 1)!) ) - 126 %2 %1 | / / \ n 2 3 4 4 (15/2 - 239/4 n + 203 n - 1129/4 n - 337/2 n ) 5 4 3 2 16 %1 (-1 + 2 n) (2210 n + 24178 n + 904 n - 5503 n + 2151 n - 315) + --- -------------------------------------------------------------------- 315 n! (n - 1)! \ / | / 3 3 6 2 | | / ((n!) ((n - 1)!) ) + 84 %2 %1 | / / \ n 2 3 4 (7/2 - 81/4 n + 141/4 n + 15/2 n ) 3 2 \ %1 (-1 + 2 n) (63 n + 21 n - 24 n + 5)| / 2 2 - 8/5 ----------------------------------------| / ((n!) ((n - 1)!) ) n! (n - 1)! / / / 2 \ \ 7 | n / 9 n\ %1 (-1 + 2 n) (5 n + 7 n - 3)| | 36 %2 %1 |4 |3/2 - ---| + 4/3 ------------------------------| | \ \ 2 / n! (n - 1)! / 9| - --------------------------------------------------------------- + 8 %2 | n! (n - 1)! / 9 9 / 9 (n!) ((n - 1)!) / %1 / %1 := (2 n - 2)! n 4 2 %1 (-1 + 2 n) %2 := ---- - --------------- 2 n! (n - 1)! and in Maple notation by ((2*n-2)!^8/n!^8/(n-1)!^8*(4^n*(511/2+38234255219/320*n^10+54208131033/128*n^9+ 122845422387/256*n^8+16145585/176*n^12+7455425931/160*n^7+16725648203/256*n^4+ 751249461/80*n^11-127978887571/640*n^6-4012381707/128*n^5+972410319/320*n^3-\ 31713828081/3520*n^2+3078273/16*n)-512/111546435*(2*n-2)!/n!/(n-1)!*(-1+2*n)*( 1048241727120*n^12+31226624575240*n^11+204201623034104*n^10+429879431852561*n^9 +276904518461427*n^8-77822821450878*n^7-122633209301358*n^6+22615990332254*n^5+ 33039275776063*n^4-4175143669772*n^3-3809693871861*n^2+83749453200*n+111546435) )-9*(1/2*4^n-2*(2*n-2)!/n!/(n-1)!*(-1+2*n))*(2*n-2)!^7/n!^7/(n-1)!^7*(4^n*(255/ 2-860131/4*n^10-26628503/6*n^9-284199207/16*n^8-557796517/32*n^7+7386361/16*n^4 +367437*n^6+82952807/16*n^5-114461335/96*n^3+296505/8*n^2+437985/8*n)+256/ 43648605*(2*n-2)!/n!/(n-1)!*(-1+2*n)*(10256252200*n^11+759125283404*n^10+ 6488988041724*n^9+13625670130371*n^8+6702304036344*n^7-2251820343192*n^6-\ 2257445336494*n^5+403059409615*n^4+402124726704*n^3-38541939273*n^2-18740302623 *n-43648605))+36*(1/2*4^n-2*(2*n-2)!/n!/(n-1)!*(-1+2*n))^2*(2*n-2)!^6/n!^6/(n-1 )!^6*(4^n*(127/2+19675/6*n^9+167623*n^8+13722379/16*n^7-3220021/32*n^4+23152269 /32*n^6-4553773/32*n^5-1593679/96*n^3+45029/2*n^2-13835/8*n)-128/109395*(2*n-2) !/n!/(n-1)!*(-1+2*n)*(105889684*n^9+1446232944*n^8+3247714950*n^7+1000233162*n^ 6-448344324*n^5-236075874*n^4+26229887*n^3+34457727*n^2-2829816*n+109395))-84*( 1/2*4^n-2*(2*n-2)!/n!/(n-1)!*(-1+2*n))^3*(2*n-2)!^5/n!^5/(n-1)!^5*(4^n*(63/2-\ 162831/28*n^7+270025/16*n^4-378665/8*n^6-598257/16*n^5-39833/16*n^3+27825/16*n^ 2-30575/56*n)+64/9009*(2*n-2)!/n!/(n-1)!*(-1+2*n)*(331300*n^8+11114630*n^7+ 30810169*n^6+3706098*n^5-3614193*n^4-5018*n^3-288419*n^2+143589*n-9009))+126*(1 /2*4^n-2*(2*n-2)!/n!/(n-1)!*(-1+2*n))^4*(2*n-2)!^4/n!^4/(n-1)!^4*(4^n*(31/2+ 43635/16*n^4+565/4*n^6+22817/8*n^5-1865*n^3+9725/16*n^2-1207/8*n)-32/231*(2*n-2 )!/n!/(n-1)!*(-1+2*n)*(25224*n^6+110447*n^5+184*n^4-15350*n^3+6203*n^2-1968*n+ 231))-126*(1/2*4^n-2*(2*n-2)!/n!/(n-1)!*(-1+2*n))^5*(2*n-2)!^3/n!^3/(n-1)!^3*(4 ^n*(15/2-239/4*n+203*n^2-1129/4*n^3-337/2*n^4)+16/315*(2*n-2)!/n!/(n-1)!*(-1+2* n)*(2210*n^5+24178*n^4+904*n^3-5503*n^2+2151*n-315))+84*(1/2*4^n-2*(2*n-2)!/n!/ (n-1)!*(-1+2*n))^6*(2*n-2)!^2/n!^2/(n-1)!^2*(4^n*(7/2-81/4*n+141/4*n^2+15/2*n^3 )-8/5*(2*n-2)!/n!/(n-1)!*(-1+2*n)*(63*n^3+21*n^2-24*n+5))-36*(1/2*4^n-2*(2*n-2) !/n!/(n-1)!*(-1+2*n))^7*(2*n-2)!/n!/(n-1)!*(4^n*(3/2-9/2*n)+4/3*(2*n-2)!/n!/(n-\ 1)!*(-1+2*n)*(5*n^2+7*n-3))+8*(1/2*4^n-2*(2*n-2)!/n!/(n-1)!*(-1+2*n))^9)/(2*n-2 )!^9*n!^9*(n-1)!^9 / 1/2 | 6978220262525 Pi this is asymptotically, where m=sqrt(n) , |- ------------------- \ 3879876 3/2 \ 546709600 Pi (7/2) (9/2) (5/2)| 27 - --------------- + 99840 Pi + 4096 Pi + 233824 Pi | m + 429 / / 4 9698065834 118057804988416 16311520 2 3\ |9216 Pi - ---------- Pi - --------------- + -------- Pi + 2792832/5 Pi | \ 1105 22309287 11 / / 3/2 5/2 1/2 26 |499596580 Pi 17280972 Pi 2048532640376919 Pi m + |--------------- + -------------- - ---------------------- \ 143 5 51731680 \ (9/2) (7/2)| 25 / 550662306003 4 - 13824 Pi - 130752 Pi | m + |- ------------ Pi - 18432 Pi / \ 24310 / 175732384 2 712542313029632 3\ 24 | (9/2) + --------- Pi - --------------- - 468384/5 Pi | m + |18720 Pi 11 8580495 / \ 3/2 5/2 33004928815 Pi (7/2) 5710239 Pi + ----------------- - 65844 Pi + ------------- 572 20 1/2\ 148991274935817939 Pi | 23 /155709788 2 3 - ------------------------| m + |--------- Pi - 5975844/5 Pi 827706880 / \ 11 9904557360619520 1759093387699 4\ 22 - ---------------- + ------------- Pi + 4608 Pi | m 22309287 17680 / and in Maple notation (-6978220262525/3879876*Pi^(1/2)-546709600/429*Pi^(3/2)+99840*Pi^(7/2)+4096*Pi^ (9/2)+233824*Pi^(5/2))*m^27+(9216*Pi^4-9698065834/1105*Pi-118057804988416/ 22309287+16311520/11*Pi^2+2792832/5*Pi^3)*m^26+(499596580/143*Pi^(3/2)+17280972 /5*Pi^(5/2)-2048532640376919/51731680*Pi^(1/2)-13824*Pi^(9/2)-130752*Pi^(7/2))* m^25+(-550662306003/24310*Pi-18432*Pi^4+175732384/11*Pi^2-712542313029632/ 8580495-468384/5*Pi^3)*m^24+(18720*Pi^(9/2)+33004928815/572*Pi^(3/2)-65844*Pi^( 7/2)+5710239/20*Pi^(5/2)-148991274935817939/827706880*Pi^(1/2))*m^23+(155709788 /11*Pi^2-5975844/5*Pi^3-9904557360619520/22309287+1759093387699/17680*Pi+4608* Pi^4)*m^22 Finally, the limit of the average divided by n^(3/2), the limit of the coeff\ icient of variation, and the limit of the scaled moment up to the, 9, -th are 14144 2 1/2 (3/2) 1/2 ----- - 48 Pi + 80 Pi 1/2 (30 - 9 Pi) 16 Pi - 50 Pi 63 [2 Pi , --------------, ---------------------, ----------------------, 1/2 3/2 2 3 Pi (40/3 - 4 Pi) (40/3 - 4 Pi) 1/2 3/2 (5/2) 105845 Pi 400 Pi 128 Pi - ------------ + --------- 63 3 --------------------------------------, 5/2 (40/3 - 4 Pi) 2 42406400 140500 3 -1600 Pi + -------- + ------ Pi - 320 Pi 9009 21 ------------------------------------------, 3 (40/3 - 4 Pi) 3/2 1/2 138620 Pi (5/2) (7/2) 67931650 Pi - ------------ + 7840 Pi + 768 Pi - -------------- 9 1287 ------------------------------------------------------------, 7/2 (40/3 - 4 Pi) 1050240225280 408404000 3 2 4 ------------- + --------- Pi - 89600/3 Pi - 15040/9 Pi - 1792 Pi / 8729721 1287 | -------------------------------------------------------------------, | 4 \ (40/3 - 4 Pi) 1/2 3/2 6978220262525 Pi 546709600 Pi (7/2) (9/2) - ------------------- - --------------- + 99840 Pi + 4096 Pi 3879876 429 \ (5/2)| / 9/2 + 233824 Pi | / (40/3 - 4 Pi) ] / / and in Maple notation [2*Pi^(1/2), 1/3*(30-9*Pi)^(1/2)/Pi^(1/2), (16*Pi^(3/2)-50*Pi^(1/2))/(40/3-4*Pi )^(3/2), (14144/63-48*Pi^2+80*Pi)/(40/3-4*Pi)^2, (128*Pi^(5/2)-105845/63*Pi^(1/ 2)+400/3*Pi^(3/2))/(40/3-4*Pi)^(5/2), (-1600*Pi^2+42406400/9009+140500/21*Pi-\ 320*Pi^3)/(40/3-4*Pi)^3, (-138620/9*Pi^(3/2)+7840*Pi^(5/2)+768*Pi^(7/2)-\ 67931650/1287*Pi^(1/2))/(40/3-4*Pi)^(7/2), (1050240225280/8729721+408404000/ 1287*Pi-89600/3*Pi^3-15040/9*Pi^2-1792*Pi^4)/(40/3-4*Pi)^4, (-6978220262525/ 3879876*Pi^(1/2)-546709600/429*Pi^(3/2)+99840*Pi^(7/2)+4096*Pi^(9/2)+233824*Pi^ (5/2))/(40/3-4*Pi)^(9/2)] And in floating-point: [3.5449077018110320546, .24704848501047098128, .70056652935965031715, 3.5603948\ 971328890447, 7.2563753582799571273, 27.685525695770609292, 90.0171829093603301\ 12, 358.80904151261251182, 1460.7011342971821191] ----------------------------------------------- This ends this article, that took, 2298.148, seconds to generate.