The Statistics of the Limiting Occupancy of the cell [1,i] in a 2-rowed Stan\ dard Young tableau of shape [n,n], as n goes to infinity, and its Asympo\ tic behavior as i goes to infinity By Shalosh B. Ekhad In this article we will discuss the limiting statistics of the occupant of t\ he cell [1,i] in a 2-rowed standard Young tableau of shape [n,n] as n g\ oes to infinity, and later as i goes to infinity Of course the occupant can be anywhere between i and 2*i-1, regardless of n.\ As n goes to infinity the expected size of the occupant of cell [1,i] is (-i) 2 4 (1 + 2 i)! 2 i + 2 - ------------------ 2 (i!) and in Maple notation 2*i+2-2*4^(-i)*(1+2*i)!/i!^2 This is asymptotically, as i goes to infinity 1/2 1/2 1/2 4 3 (1/i) 7 (1/i) 9 (1/i) 2 + 2 i - -------------- - ---------- + ---------- - ------------ 1/2 1/2 1/2 1/2 1/2 2 Pi (1/i) 2 Pi 32 Pi i 256 Pi i 1/2 1/2 1/2 59 (1/i) 483 (1/i) (1/i) - ------------- + -------------- + O(--------) 1/2 3 1/2 4 5 8192 Pi i 65536 Pi i i and in Maple notation 2+2*i-4/Pi^(1/2)/(1/i)^(1/2)-3/2/Pi^(1/2)*(1/i)^(1/2)+7/32/Pi^(1/2)/i*(1/i)^(1/ 2)-9/256/Pi^(1/2)/i^2*(1/i)^(1/2)-59/8192/Pi^(1/2)/i^3*(1/i)^(1/2)+483/65536/Pi ^(1/2)/i^4*(1/i)^(1/2)+O(1/i^5*(1/i)^(1/2)) the variance of the occupant of cell [1,i] is (-i) 2 (-i) 4 16 ((1 + 2 i)!) 2 4 (1 + 2 i)! - ---------------------- - ------------------ + 6 i + 6 4 2 (i!) (i!) and in Maple notation -4*16^(-i)*(1+2*i)!^2/i!^4-2*4^(-i)*(1+2*i)!/i!^2+6*i+6 This is asymptotically, as i goes to infinity 1/2 1/2 2 (3 Pi - 8) i 4 12 3 (1/i) 1 7 (1/i) 6 + -------------- - -------------- - ---- - ---------- - ------ + ---------- Pi 1/2 1/2 Pi 1/2 2 Pi i 1/2 Pi (1/i) 2 Pi 32 Pi i 1/2 1/2 3 9 (1/i) 27 59 (1/i) 27 + ------- - ------------ - --------- - ------------- + --------- 2 1/2 2 3 1/2 3 4 8 Pi i 256 Pi i 128 Pi i 8192 Pi i 512 Pi i 1/2 483 (1/i) 1 + -------------- + O(----) 1/2 4 5 65536 Pi i i and in Maple notation 6+2*(3*Pi-8)/Pi*i-4/Pi^(1/2)/(1/i)^(1/2)-12/Pi-3/2/Pi^(1/2)*(1/i)^(1/2)-1/2/Pi/ i+7/32/Pi^(1/2)/i*(1/i)^(1/2)+3/8/Pi/i^2-9/256/Pi^(1/2)/i^2*(1/i)^(1/2)-27/128/ Pi/i^3-59/8192/Pi^(1/2)/i^3*(1/i)^(1/2)+27/512/Pi/i^4+483/65536/Pi^(1/2)/i^4*(1 /i)^(1/2)+O(1/i^5) The , 3, -th moment about the mean is -16*64^(-i)*(1+2*i)!^3/i!^6-12*16^(-i)*(1+2*i)!^2/i!^4+(20*i+10)*4^(-i)*(1+2*i) !/i!^2+18+18*i This is asymptotically, as i goes to infinity 8 (5 Pi - 16) i 6 (3 Pi - 8) i 35 Pi - 144 36 18 + --------------- + -------------- + -------------- - ---- 3/2 1/2 Pi 3/2 1/2 Pi Pi (1/i) Pi (1/i) 1/2 1/2 (85 Pi - 528) (1/i) 3 5 (19 Pi - 144) (1/i) 9 + ---------------------- - ------ - ------------------------ + ------- 3/2 2 Pi i 3/2 2 16 Pi 128 i Pi 8 Pi i 1/2 1/2 (1015 Pi - 5808) (1/i) 81 (1235 Pi + 11376) (1/i) + ------------------------- - --------- - -------------------------- 2 3/2 3 3 3/2 4096 i Pi 128 Pi i 32768 i Pi 1/2 81 (1/i) + --------- + O(--------) 4 4 512 Pi i i and in Maple notation 18+8*(5*Pi-16)/Pi^(3/2)*i/(1/i)^(1/2)+6*(3*Pi-8)/Pi*i+(35*Pi-144)/Pi^(3/2)/(1/i )^(1/2)-36/Pi+1/16*(85*Pi-528)*(1/i)^(1/2)/Pi^(3/2)-3/2/Pi/i-5/128*(19*Pi-144)/ i*(1/i)^(1/2)/Pi^(3/2)+9/8/Pi/i^2+1/4096*(1015*Pi-5808)/i^2*(1/i)^(1/2)/Pi^(3/2 )-81/128/Pi/i^3-1/32768*(1235*Pi+11376)/i^3*(1/i)^(1/2)/Pi^(3/2)+81/512/Pi/i^4+ O(1/i^4*(1/i)^(1/2)) The , 4, -th moment about the mean is -48*256^(-i)*(1+2*i)!^4/i!^8-48*64^(-i)*(1+2*i)!^3/i!^6+2^(-4*i+4)*(1+2*i)!^2*( i-4)/i!^4+(48*i-2)*4^(-i)*(1+2*i)!/i!^2+60*i^2+222*i+162 This is asymptotically, as i goes to infinity 2 2 4 (15 Pi + 16 Pi - 192) i 96 (Pi - 4) i 162 + --------------------------- + -------------- 2 3/2 1/2 Pi Pi (1/i) 2 2 (111 Pi - 104 Pi - 576) i 32 Pi - 432 480 190 + ---------------------------- + -------------- - --- - --- 2 3/2 1/2 2 Pi Pi Pi (1/i) Pi 1/2 1/2 9 (3 Pi + 44) (1/i) 19 (17 Pi + 270) (1/i) - ---------------------- - ------ + ---------------------- 3/2 2 Pi i 3/2 4 Pi 16 i Pi 1/2 3 (73 Pi + 64) 3 (47 Pi - 1452) (1/i) 9 (51 Pi + 128) + -------------- + ------------------------- - --------------- 2 2 2 3/2 3 2 32 i Pi 1024 i Pi 128 i Pi 1/2 (377 Pi + 2133) (1/i) 1 - ------------------------ + O(----) 3 3/2 4 2048 i Pi i and in Maple notation 162+4*(15*Pi^2+16*Pi-192)/Pi^2*i^2+96*(Pi-4)/Pi^(3/2)*i/(1/i)^(1/2)+2*(111*Pi^2 -104*Pi-576)/Pi^2*i+(32*Pi-432)/Pi^(3/2)/(1/i)^(1/2)-480/Pi^2-190/Pi-9/4*(3*Pi+ 44)*(1/i)^(1/2)/Pi^(3/2)-19/2/Pi/i+1/16*(17*Pi+270)/i*(1/i)^(1/2)/Pi^(3/2)+3/32 *(73*Pi+64)/i^2/Pi^2+3/1024*(47*Pi-1452)/i^2*(1/i)^(1/2)/Pi^(3/2)-9/128*(51*Pi+ 128)/i^3/Pi^2-1/2048*(377*Pi+2133)/i^3*(1/i)^(1/2)/Pi^(3/2)+O(1/i^4) The , 5, -th moment about the mean is -128*1024^(-i)*(1+2*i)!^5/i!^10-160*256^(-i)*(1+2*i)!^4/i!^8+(-160*i-560)*64^(- i)*(1+2*i)!^3/i!^6+(-240*i-740)*16^(-i)*(1+2*i)!^2/i!^4+(408*i^2+1052*i+298)*4^ (-i)*(1+2*i)!/i!^2+600*i^2+1890*i+1290 This is asymptotically, as i goes to infinity 2 2 2 2 16 (51 Pi - 80 Pi - 256) i 40 (15 Pi - 24 Pi - 64) i 1290 + ---------------------------- + --------------------------- 5/2 1/2 2 Pi (1/i) Pi 2 2 10 (241 Pi - 592 Pi - 768) i 10 (189 Pi - 368 Pi - 384) i + ----------------------------- + ----------------------------- 5/2 1/2 2 Pi (1/i) Pi 2 10723 Pi - 42960 Pi - 37120 2250 1600 + ---------------------------- - ---- - ---- 5/2 1/2 Pi 2 8 Pi (1/i) Pi 2 1/2 (7399 Pi - 70320 Pi - 42240) (1/i) 70 + -------------------------------------- - ---- 5/2 Pi i 64 Pi 2 1/2 (25871 Pi - 374160 Pi - 226560) (1/i) 5 (363 Pi + 128) - ----------------------------------------- + ---------------- 5/2 2 2 2048 i Pi 32 i Pi 2 1/2 (123259 Pi - 870000 Pi - 1016064) (1/i) 15 (153 Pi + 128) + ------------------------------------------- - ----------------- 2 5/2 3 2 16384 i Pi 64 i Pi 1/2 (1/i) + O(--------) 3 i and in Maple notation 1290+16*(51*Pi^2-80*Pi-256)/Pi^(5/2)*i^2/(1/i)^(1/2)+40*(15*Pi^2-24*Pi-64)/Pi^2 *i^2+10*(241*Pi^2-592*Pi-768)/Pi^(5/2)*i/(1/i)^(1/2)+10*(189*Pi^2-368*Pi-384)/ Pi^2*i+1/8*(10723*Pi^2-42960*Pi-37120)/Pi^(5/2)/(1/i)^(1/2)-2250/Pi-1600/Pi^2+1 /64*(7399*Pi^2-70320*Pi-42240)*(1/i)^(1/2)/Pi^(5/2)-70/Pi/i-1/2048*(25871*Pi^2-\ 374160*Pi-226560)/i*(1/i)^(1/2)/Pi^(5/2)+5/32*(363*Pi+128)/i^2/Pi^2+1/16384*( 123259*Pi^2-870000*Pi-1016064)/i^2*(1/i)^(1/2)/Pi^(5/2)-15/64*(153*Pi+128)/i^3/ Pi^2+O(1/i^3*(1/i)^(1/2)) The , 6, -th moment about the mean is -320*4096^(-i)*(1+2*i)!^6/i!^12-480*1024^(-i)*(1+2*i)!^5/i!^10+(-1120*i-2720)* 256^(-i)*(1+2*i)!^4/i!^8+(-2880*i-5880)*64^(-i)*(1+2*i)!^3/i!^6+(1296*i^2-696*i -6144)*16^(-i)*(1+2*i)!^2/i!^4+(4320*i^2+9720*i+2518)*4^(-i)*(1+2*i)!/i!^2+840* i^3+8940*i^2+21126*i+13026 This is asymptotically, as i goes to infinity 3 2 3 2 2 8 (105 Pi + 648 Pi - 2240 Pi - 2560) i 960 (9 Pi - 24 Pi - 16) i 13026 + ----------------------------------------- + --------------------------- 3 5/2 1/2 Pi Pi (1/i) 3 2 2 4 (2235 Pi + 276 Pi - 17600 Pi - 11520) i + -------------------------------------------- 3 Pi 2 120 (189 Pi - 608 Pi - 240) i + ------------------------------ 5/2 1/2 Pi (1/i) 3 2 2 (10563 Pi - 13251 Pi - 38240 Pi - 18240) i + ---------------------------------------------- 3 Pi 2 23707 Pi - 117720 Pi - 34800 27200 10080 37281 + ----------------------------- - ----- - ----- - ----- 5/2 1/2 2 3 2 Pi 2 Pi (1/i) Pi Pi 2 1/2 2 3 (4807 Pi - 59280 Pi - 13200) (1/i) -20301 Pi + 4480 Pi + 6720 + ---------------------------------------- + --------------------------- 5/2 3 16 Pi 32 i Pi 2 1/2 2 (45563 Pi - 927720 Pi - 212400) (1/i) 66843 Pi + 16640 Pi - 20160 - ----------------------------------------- + ---------------------------- 5/2 2 3 512 i Pi 128 i Pi 2 1/2 9 (28829 Pi - 265600 Pi - 105840) (1/i) 1 + ------------------------------------------- + O(----) 2 5/2 3 4096 i Pi i and in Maple notation 13026+8*(105*Pi^3+648*Pi^2-2240*Pi-2560)/Pi^3*i^3+960*(9*Pi^2-24*Pi-16)/Pi^(5/2 )*i^2/(1/i)^(1/2)+4*(2235*Pi^3+276*Pi^2-17600*Pi-11520)/Pi^3*i^2+120*(189*Pi^2-\ 608*Pi-240)/Pi^(5/2)*i/(1/i)^(1/2)+2*(10563*Pi^3-13251*Pi^2-38240*Pi-18240)/Pi^ 3*i+1/2*(23707*Pi^2-117720*Pi-34800)/Pi^(5/2)/(1/i)^(1/2)-27200/Pi^2-10080/Pi^3 -37281/2/Pi+3/16*(4807*Pi^2-59280*Pi-13200)*(1/i)^(1/2)/Pi^(5/2)+1/32*(-20301* Pi^2+4480*Pi+6720)/i/Pi^3-1/512*(45563*Pi^2-927720*Pi-212400)/i*(1/i)^(1/2)/Pi^ (5/2)+1/128*(66843*Pi^2+16640*Pi-20160)/i^2/Pi^3+9/4096*(28829*Pi^2-265600*Pi-\ 105840)/i^2*(1/i)^(1/2)/Pi^(5/2)+O(1/i^3) The , 7, -th moment about the mean is -768*16384^(-i)*(1+2*i)!^7/i!^14-1344*4096^(-i)*(1+2*i)!^6/i!^12+(-4928*i-10528 )*1024^(-i)*(1+2*i)!^5/i!^10+(-16800*i-30800)*256^(-i)*(1+2*i)!^4/i!^8+(672*i^2 -35952*i-65688)*64^(-i)*(1+2*i)!^3/i!^6+(10080*i^2-22680*i-73108)*16^(-i)*(1+2* i)!^2/i!^4+(8688*i^3+75240*i^2+128900*i+34018)*4^(-i)*(1+2*i)!/i!^2+17640*i^3+ 126840*i^2+257418*i+148218 This is asymptotically, as i goes to infinity 3 2 3 32 (543 Pi + 168 Pi - 4928 Pi - 3072) i 148218 + ------------------------------------------ 7/2 1/2 Pi (1/i) 3 2 3 168 (105 Pi + 240 Pi - 1600 Pi - 512) i + ------------------------------------------ 3 Pi 3 2 2 28 (5607 Pi - 10056 Pi - 22592 Pi - 9216) i + ---------------------------------------------- 7/2 1/2 Pi (1/i) 3 2 2 56 (2265 Pi - 1080 Pi - 16000 Pi - 3456) i + --------------------------------------------- 3 Pi 3 2 7 (179017 Pi - 484392 Pi - 463040 Pi - 144384) i + -------------------------------------------------- 7/2 1/2 4 Pi (1/i) 3 2 14 (18387 Pi - 25658 Pi - 64800 Pi - 10944) i + ----------------------------------------------- 3 Pi 3 2 5012299 Pi - 21298536 Pi - 13025600 Pi - 3290112 223104 308000 + -------------------------------------------------- - ------ - ------ 7/2 1/2 Pi 2 32 Pi (1/i) Pi 3 2 1/2 42336 (13075381 Pi - 125729352 Pi - 51226560 Pi - 9913344) (1/i) - ----- + --------------------------------------------------------------- 3 7/2 Pi 1024 Pi 2 7 (14813 Pi - 4800 Pi - 2016) - ------------------------------ 3 16 i Pi 3 2 1/2 7 (1423031 Pi - 23314248 Pi - 7854144 Pi - 599040) (1/i) - ------------------------------------------------------------- 7/2 8192 i Pi 2 1/2 7 (25257 Pi + 3200 Pi - 3024) (1/i) + ------------------------------ + O(--------) 2 3 2 32 i Pi i and in Maple notation 148218+32*(543*Pi^3+168*Pi^2-4928*Pi-3072)/Pi^(7/2)*i^3/(1/i)^(1/2)+168*(105*Pi ^3+240*Pi^2-1600*Pi-512)/Pi^3*i^3+28*(5607*Pi^3-10056*Pi^2-22592*Pi-9216)/Pi^(7 /2)*i^2/(1/i)^(1/2)+56*(2265*Pi^3-1080*Pi^2-16000*Pi-3456)/Pi^3*i^2+7/4*(179017 *Pi^3-484392*Pi^2-463040*Pi-144384)/Pi^(7/2)*i/(1/i)^(1/2)+14*(18387*Pi^3-25658 *Pi^2-64800*Pi-10944)/Pi^3*i+1/32*(5012299*Pi^3-21298536*Pi^2-13025600*Pi-\ 3290112)/Pi^(7/2)/(1/i)^(1/2)-223104/Pi-308000/Pi^2-42336/Pi^3+1/1024*(13075381 *Pi^3-125729352*Pi^2-51226560*Pi-9913344)*(1/i)^(1/2)/Pi^(7/2)-7/16*(14813*Pi^2 -4800*Pi-2016)/i/Pi^3-7/8192*(1423031*Pi^3-23314248*Pi^2-7854144*Pi-599040)/i*( 1/i)^(1/2)/Pi^(7/2)+7/32*(25257*Pi^2+3200*Pi-3024)/i^2/Pi^3+O(1/i^2*(1/i)^(1/2) ) The , 8, -th moment about the mean is -1792*65536^(-i)*(1+2*i)!^8/i!^16-3584*16384^(-i)*(1+2*i)!^7/i!^14-17920*4096^( -i)*(1+2*i)!^6*(i+2)/i!^12+(-75264*i-131264)*1024^(-i)*(1+2*i)!^5/i!^10+(-18816 *i^2-274624*i-410816)*256^(-i)*(1+2*i)!^4/i!^8+(-53760*i^2-604800*i-873824)*64^ (-i)*(1+2*i)!^3/i!^6+(44928*i^3+202560*i^2-303712*i-914624)*16^(-i)*(1+2*i)!^2/ i!^4+(196224*i^3+1182720*i^2+1770048*i+463342)*4^(-i)*(1+2*i)!/i!^2+15120*i^4+ 374640*i^3+1971900*i^2+3520782*i+1908402 This is asymptotically, as i goes to infinity 4 3 2 4 16 (945 Pi + 11232 Pi - 18816 Pi - 71680 Pi - 28672) i 1908402 + ---------------------------------------------------------- 4 Pi 3 2 3 1792 (219 Pi - 240 Pi - 1344 Pi - 256) i + ------------------------------------------- 7/2 1/2 Pi (1/i) 4 3 2 3 16 (23415 Pi + 59064 Pi - 302848 Pi - 304640 Pi - 86016) i + -------------------------------------------------------------- 4 Pi 3 2 2 224 (11217 Pi - 23760 Pi - 38912 Pi - 5376) i + ------------------------------------------------ 7/2 1/2 Pi (1/i) 4 3 2 2 28 (70425 Pi - 21484 Pi - 476864 Pi - 257280 Pi - 57344) i + -------------------------------------------------------------- 4 Pi 3 2 14 (314691 Pi - 896048 Pi - 757440 Pi - 84224) i + -------------------------------------------------- 7/2 1/2 Pi (1/i) 4 3 2 2 (1760391 Pi - 2274262 Pi - 6302912 Pi - 2325120 Pi - 430080) i + ------------------------------------------------------------------- 4 Pi 3 2 8513237 Pi - 36371664 Pi - 20585600 Pi - 1919232 11193827 1117200 + -------------------------------------------------- - -------- - ------- 7/2 1/2 4 Pi 3 4 Pi (1/i) Pi 172032 4105808 - ------ - ------- 4 2 Pi Pi 3 2 1/2 (22451727 Pi - 204083376 Pi - 78308160 Pi - 5782784) (1/i) + --------------------------------------------------------------- 7/2 128 Pi 2 -1212247 Pi + 492800 Pi + 235200 + --------------------------------- 3 16 i Pi 3 2 1/2 7 (2344657 Pi - 36928080 Pi - 11261184 Pi - 349440) (1/i) 1 - -------------------------------------------------------------- + O(----) 7/2 2 1024 i Pi i and in Maple notation 1908402+16*(945*Pi^4+11232*Pi^3-18816*Pi^2-71680*Pi-28672)/Pi^4*i^4+1792*(219* Pi^3-240*Pi^2-1344*Pi-256)/Pi^(7/2)*i^3/(1/i)^(1/2)+16*(23415*Pi^4+59064*Pi^3-\ 302848*Pi^2-304640*Pi-86016)/Pi^4*i^3+224*(11217*Pi^3-23760*Pi^2-38912*Pi-5376) /Pi^(7/2)*i^2/(1/i)^(1/2)+28*(70425*Pi^4-21484*Pi^3-476864*Pi^2-257280*Pi-57344 )/Pi^4*i^2+14*(314691*Pi^3-896048*Pi^2-757440*Pi-84224)/Pi^(7/2)*i/(1/i)^(1/2)+ 2*(1760391*Pi^4-2274262*Pi^3-6302912*Pi^2-2325120*Pi-430080)/Pi^4*i+1/4*( 8513237*Pi^3-36371664*Pi^2-20585600*Pi-1919232)/Pi^(7/2)/(1/i)^(1/2)-11193827/4 /Pi-1117200/Pi^3-172032/Pi^4-4105808/Pi^2+1/128*(22451727*Pi^3-204083376*Pi^2-\ 78308160*Pi-5782784)*(1/i)^(1/2)/Pi^(7/2)+1/16*(-1212247*Pi^2+492800*Pi+235200) /i/Pi^3-7/1024*(2344657*Pi^3-36928080*Pi^2-11261184*Pi-349440)/i*(1/i)^(1/2)/Pi ^(7/2)+O(1/i^2) The , 9, -th moment about the mean is -4096*262144^(-i)*(1+2*i)!^9/i!^18-9216*65536^(-i)*(1+2*i)!^8/i!^16+(-58368*i-\ 112128)*16384^(-i)*(1+2*i)!^7/i!^14+(-290304*i-491904)*4096^(-i)*(1+2*i)!^6/i!^ 12-4032*1024^(-i)*(1+2*i)!^5*(36*i^2+362*i+499)/i!^10+(-725760*i^2-4898880*i-\ 6109824)*256^(-i)*(1+2*i)!^4/i!^8+(122112*i^3-1180800*i^2-9831744*i-12608352)* 64^(-i)*(1+2*i)!^3/i!^6+(991872*i^3+3024000*i^2-5207328*i-13003236)*16^(-i)*(1+ 2*i)!^2/i!^4+(210720*i^4+4586976*i^3+20350968*i^2+27081452*i+7017274)*4^(-i)*(1 +2*i)!/i!^2+544320*i^4+7665840*i^3+32853240*i^2+53065170*i+27333450 This is asymptotically, as i goes to infinity 4 3 2 4 64 (6585 Pi + 15264 Pi - 72576 Pi - 116736 Pi - 32768) i 27333450 + ------------------------------------------------------------ 9/2 1/2 Pi (1/i) 4 3 2 4 576 (945 Pi + 6888 Pi - 20160 Pi - 32256 Pi - 4096) i + --------------------------------------------------------- 4 Pi 4 3 2 3 24 (388833 Pi - 347808 Pi - 2308992 Pi - 1415168 Pi - 294912) i + ------------------------------------------------------------------- 9/2 1/2 Pi (1/i) 4 3 2 3 144 (53235 Pi + 104664 Pi - 665280 Pi - 508928 Pi - 49152) i + ---------------------------------------------------------------- + 3 4 Pi 4 3 2 2 (29412747 Pi - 59352864 Pi - 104813184 Pi - 37918720 Pi - 6389760) i / 9/2 1/2 / (2 Pi (1/i) ) / 4 3 2 2 168 (195555 Pi - 69246 Pi - 1324928 Pi - 618624 Pi - 49152) i + ----------------------------------------------------------------- + ( 4 Pi 4 3 2 1102850137 Pi - 3069293472 Pi - 2790023040 Pi - 715266048 Pi - 101744640 / 9/2 1/2 ) i / (16 Pi (1/i) ) / 4 3 2 18 (2948065 Pi - 3741662 Pi - 10868032 Pi - 3623424 Pi - 245760) i + --------------------------------------------------------------------- + 4 Pi 4 3 2 (16486983235 Pi - 68263522848 Pi - 41131198080 Pi - 8062408704 Pi / 9/2 1/2 61007520 15304464 159567273 - 982351872) / (512 Pi (1/i) ) - -------- - -------- - --------- / 2 3 4 Pi Pi Pi 884736 4 3 2 - ------ + (10954322269 Pi - 92775424032 Pi - 37613890944 Pi 4 Pi 1/2 / 9/2 - 5630048256 Pi - 621969408) (1/i) / (4096 Pi ) / 2 1/2 9 (1761467 Pi - 846720 Pi - 319872) (1/i) - ------------------------------------ + O(--------) 3 i 16 i Pi and in Maple notation 27333450+64*(6585*Pi^4+15264*Pi^3-72576*Pi^2-116736*Pi-32768)/Pi^(9/2)*i^4/(1/i )^(1/2)+576*(945*Pi^4+6888*Pi^3-20160*Pi^2-32256*Pi-4096)/Pi^4*i^4+24*(388833* Pi^4-347808*Pi^3-2308992*Pi^2-1415168*Pi-294912)/Pi^(9/2)*i^3/(1/i)^(1/2)+144*( 53235*Pi^4+104664*Pi^3-665280*Pi^2-508928*Pi-49152)/Pi^4*i^3+3/2*(29412747*Pi^4 -59352864*Pi^3-104813184*Pi^2-37918720*Pi-6389760)/Pi^(9/2)*i^2/(1/i)^(1/2)+168 *(195555*Pi^4-69246*Pi^3-1324928*Pi^2-618624*Pi-49152)/Pi^4*i^2+1/16*( 1102850137*Pi^4-3069293472*Pi^3-2790023040*Pi^2-715266048*Pi-101744640)/Pi^(9/2 )*i/(1/i)^(1/2)+18*(2948065*Pi^4-3741662*Pi^3-10868032*Pi^2-3623424*Pi-245760)/ Pi^4*i+1/512*(16486983235*Pi^4-68263522848*Pi^3-41131198080*Pi^2-8062408704*Pi-\ 982351872)/Pi^(9/2)/(1/i)^(1/2)-61007520/Pi^2-15304464/Pi^3-159567273/4/Pi-\ 884736/Pi^4+1/4096*(10954322269*Pi^4-92775424032*Pi^3-37613890944*Pi^2-\ 5630048256*Pi-621969408)*(1/i)^(1/2)/Pi^(9/2)-9/16*(1761467*Pi^2-846720*Pi-\ 319872)/i/Pi^3+O(1/i*(1/i)^(1/2)) The , 10, -th moment about the mean is -9216*1048576^(-i)*(1+2*i)!^10/i!^20-23040*262144^(-i)*(1+2*i)!^9/i!^18+(-\ 176640*i-330240)*65536^(-i)*(1+2*i)!^8/i!^16+(-1013760*i-1685760)*16384^(-i)*(1 +2*i)!^7/i!^14+(-741888*i^2-6435072*i-8483328)*4096^(-i)*(1+2*i)!^6/i!^12-6720* 1024^(-i)*(1+2*i)!^5*(720*i^2+4212*i+4933)/i!^10+(-126720*i^3-17884800*i^2-\ 89206080*i-98644800)*256^(-i)*(1+2*i)!^4/i!^8+1451520*(i^3-190/9*i^2-121*i-\ 186275/1344)*64^(-i)*(1+2*i)!^3/i!^6+(1492800*i^4+24304320*i^3+52077360*i^2-\ 92111720*i-203166880)*16^(-i)*(1+2*i)!^2/i!^4+(8121600*i^4+102075840*i^3+ 371037600*i^2+450743880*i+115449478)*4^(-i)*(1+2*i)!/i!^2+332640*i^5+16329600*i ^4+160849080*i^3+588838380*i^2+875206806*i+431220546 This is asymptotically, as i goes to infinity 431220546 + 5 4 3 2 5 96 (3465 Pi + 62200 Pi - 21120 Pi - 494592 Pi - 471040 Pi - 98304) i ------------------------------------------------------------------------- 5 Pi 4 3 2 4 23040 (705 Pi + 504 Pi - 6720 Pi - 5632 Pi - 512) i + ------------------------------------------------------- + 960 9/2 1/2 Pi (1/i) 5 4 3 2 4 (17010 Pi + 105933 Pi - 301248 Pi - 540288 Pi - 229376 Pi - 36864) i / 5 / Pi / 4 3 2 3 960 (219003 Pi - 241752 Pi - 1245888 Pi - 579584 Pi - 41472) i + ------------------------------------------------------------------ + 24 9/2 1/2 Pi (1/i) 5 4 3 2 (6702045 Pi + 11725375 Pi - 77408320 Pi - 64756608 Pi - 17162240 Pi 3 / 5 - 2273280) i / Pi + / 4 3 2 2 60 (13629063 Pi - 27964440 Pi - 48907712 Pi - 14999040 Pi - 898560) i ------------------------------------------------------------------------- 9/2 1/2 Pi (1/i) 5 4 3 2 + 2 (294419190 Pi - 104658355 Pi - 1949055360 Pi - 989283456 Pi 2 / 5 - 190341120 Pi - 21565440) i / Pi + 15 / 4 3 2 (155832557 Pi - 433844424 Pi - 405330240 Pi - 92044288 Pi - 4769280) i / 9/2 1/2 5 4 / (2 Pi (1/i) ) + (7001654448 Pi - 8677542545 Pi / 3 2 / - 26076161280 Pi - 9354538368 Pi - 1403781120 Pi - 138424320) i / (8 / 5 4 3 2 Pi ) + (33932683319 Pi - 138363924600 Pi - 85980350400 Pi / 9/2 1/2 984236160 - 15261442560 Pi - 690716160) / (64 Pi (1/i) ) - --------- / 2 Pi 31703040 19988315675 263366964 2775168 4 - -------- - ----------- - --------- - ------- + 3 (7534386609 Pi 4 32 Pi 3 5 Pi Pi Pi 3 2 1/2 - 60734372040 Pi - 25396929600 Pi - 3511316480 Pi - 145774080) (1/i) / 9/2 / (512 Pi ) + O(1/i) / and in Maple notation 431220546+96*(3465*Pi^5+62200*Pi^4-21120*Pi^3-494592*Pi^2-471040*Pi-98304)/Pi^5 *i^5+23040*(705*Pi^4+504*Pi^3-6720*Pi^2-5632*Pi-512)/Pi^(9/2)*i^4/(1/i)^(1/2)+ 960*(17010*Pi^5+105933*Pi^4-301248*Pi^3-540288*Pi^2-229376*Pi-36864)/Pi^5*i^4+ 960*(219003*Pi^4-241752*Pi^3-1245888*Pi^2-579584*Pi-41472)/Pi^(9/2)*i^3/(1/i)^( 1/2)+24*(6702045*Pi^5+11725375*Pi^4-77408320*Pi^3-64756608*Pi^2-17162240*Pi-\ 2273280)/Pi^5*i^3+60*(13629063*Pi^4-27964440*Pi^3-48907712*Pi^2-14999040*Pi-\ 898560)/Pi^(9/2)*i^2/(1/i)^(1/2)+2*(294419190*Pi^5-104658355*Pi^4-1949055360*Pi ^3-989283456*Pi^2-190341120*Pi-21565440)/Pi^5*i^2+15/2*(155832557*Pi^4-\ 433844424*Pi^3-405330240*Pi^2-92044288*Pi-4769280)/Pi^(9/2)*i/(1/i)^(1/2)+1/8*( 7001654448*Pi^5-8677542545*Pi^4-26076161280*Pi^3-9354538368*Pi^2-1403781120*Pi-\ 138424320)/Pi^5*i+1/64*(33932683319*Pi^4-138363924600*Pi^3-85980350400*Pi^2-\ 15261442560*Pi-690716160)/Pi^(9/2)/(1/i)^(1/2)-984236160/Pi^2-31703040/Pi^4-\ 19988315675/32/Pi-263366964/Pi^3-2775168/Pi^5+3/512*(7534386609*Pi^4-\ 60734372040*Pi^3-25396929600*Pi^2-3511316480*Pi-145774080)*(1/i)^(1/2)/Pi^(9/2) +O(1/i) The limiting scaled moments about the mean, as i goes to infinity are 1/2 2 2 1/2 2 (5 Pi - 16) 2 15 Pi + 16 Pi - 192 2 (51 Pi - 80 Pi - 256) 2 [0, 1, ------------------, --------------------, -----------------------------, 3/2 2 5/2 (3 Pi - 8) (3 Pi - 8) (3 Pi - 8) 3 2 105 Pi + 648 Pi - 2240 Pi - 2560 ----------------------------------, 3 (3 Pi - 8) 3 2 1/2 2 (543 Pi + 168 Pi - 4928 Pi - 3072) 2 -------------------------------------------, 7/2 (3 Pi - 8) 4 3 2 945 Pi + 11232 Pi - 18816 Pi - 71680 Pi - 28672 --------------------------------------------------, 4 (3 Pi - 8) 4 3 2 1/2 2 (6585 Pi + 15264 Pi - 72576 Pi - 116736 Pi - 32768) 2 -------------------------------------------------------------, 9/2 (3 Pi - 8) 5 4 3 2 3 (3465 Pi + 62200 Pi - 21120 Pi - 494592 Pi - 471040 Pi - 98304) ---------------------------------------------------------------------] 5 (3 Pi - 8) and in Maple notation [0, 1, 2*(5*Pi-16)*2^(1/2)/(3*Pi-8)^(3/2), (15*Pi^2+16*Pi-192)/(3*Pi-8)^2, 2*( 51*Pi^2-80*Pi-256)*2^(1/2)/(3*Pi-8)^(5/2), (105*Pi^3+648*Pi^2-2240*Pi-2560)/(3* Pi-8)^3, 2*(543*Pi^3+168*Pi^2-4928*Pi-3072)*2^(1/2)/(3*Pi-8)^(7/2), (945*Pi^4+ 11232*Pi^3-18816*Pi^2-71680*Pi-28672)/(3*Pi-8)^4, 2*(6585*Pi^4+15264*Pi^3-72576 *Pi^2-116736*Pi-32768)*2^(1/2)/(3*Pi-8)^(9/2), 3*(3465*Pi^5+62200*Pi^4-21120*Pi ^3-494592*Pi^2-471040*Pi-98304)/(3*Pi-8)^5] and in floating point [0., 1., -.4856928234, 3.108163850, -4.642979574, 18.66866547, -48.55583418, 181.0912166, -622.5717906, 2454.604104] In more detail, the asymptotics is 1/2 1/2 1/2 3/2 (10 Pi - 32) 2 3 2 (9 Pi - 28) (1/i) Pi [0, 1, ----------------- + --------------------------------- 3/2 5/2 (3 Pi - 8) 2 (3 Pi - 8) 1/2 2 2 Pi (63 Pi - 708 Pi + 1600) - -------------------------------- 7/2 4 (3 Pi - 8) i 1/2 3 2 1/2 3/2 2 (243 Pi - 1485 Pi + 1676 Pi + 1856) (1/i) Pi - --------------------------------------------------------- 9/2 4 i (3 Pi - 8) 1/2 4 3 2 2 Pi (9801 Pi - 176904 Pi + 961776 Pi - 1991424 Pi + 1294336) + ------------------------------------------------------------------- + 11/2 2 64 (3 Pi - 8) i 1/2 5 4 3 2 2 (6561 Pi - 56619 Pi + 81261 Pi + 480756 Pi - 1591936 Pi + 1244160) 1/2 3/2 / 2 13/2 1/2 6 (1/i) Pi / (16 i (3 Pi - 8) ) - 2 Pi (741393 Pi / 5 4 3 2 - 18719748 Pi + 161472528 Pi - 655116096 Pi + 1345827840 Pi / 15/2 3 1/2 - 1330249728 Pi + 495976448) / (512 (3 Pi - 8) i ) - 2 ( / 7 6 5 4 3 393660 Pi - 4223097 Pi + 3911652 Pi + 126111276 Pi - 707568336 Pi 2 1/2 3/2 / 3 + 1633140480 Pi - 1752485888 Pi + 715849728) (1/i) Pi / (128 i / 17/2 1/2 8 7 6 (3 Pi - 8) ) + 2 Pi (218028591 Pi - 7190856000 Pi + 85562508384 Pi 5 4 3 - 518519847168 Pi + 1798599700224 Pi - 3693560082432 Pi 2 / + 4397326663680 Pi - 2774481764352 Pi + 708132732928) / (16384 / 19/2 4 1/2 9 8 7 (3 Pi - 8) i ) + 2 (12400290 Pi - 155082357 Pi - 61583004 Pi 6 5 4 3 + 11428086852 Pi - 87461955630 Pi + 327527209928 Pi - 703565263360 Pi 2 1/2 3/2 / + 878291914752 Pi - 591057649664 Pi + 165381931008) (1/i) Pi / ( / 4 21/2 1/2 10 9 512 i (3 Pi - 8) ) - 2 Pi (15653240361 Pi - 647718165036 Pi 8 7 6 + 9904673926944 Pi - 80251272360576 Pi + 393517933012224 Pi 5 4 3 - 1236454635568128 Pi + 2534737216389120 Pi - 3351731383566336 Pi 2 / + 2731484343435264 Pi - 1232740439556096 Pi + 231498737254400) / ( / 23/2 5 1/2 11 10 131072 (3 Pi - 8) i ) - 2 (1607077584 Pi - 22608385875 Pi 9 8 7 - 56053378620 Pi + 3336117562008 Pi - 32132432146608 Pi 6 5 4 + 165281284460880 Pi - 530461778010816 Pi + 1110874084613120 Pi 3 2 - 1517401615269888 Pi + 1300815649701888 Pi - 633420363333632 Pi 1/2 3/2 / 5 25/2 1 + 133093184765952) (1/i) Pi / (8192 i (3 Pi - 8) ) + O(----), / 6 i 2 3/2 1/2 15 Pi + 16 Pi - 192 (132 Pi - 416) Pi (1/i) -------------------- + ----------------------------- 2 3 (3 Pi - 8) (3 Pi - 8) 3 2 Pi (459 Pi - 3384 Pi + 10176 Pi - 12800) + ------------------------------------------ 4 2 (3 Pi - 8) i 3/2 3 2 1/2 Pi (999 Pi - 7656 Pi + 13376 Pi + 2560) (1/i) - ----------------------------------------------------- - 5 2 i (3 Pi - 8) 5 4 3 2 Pi (4131 Pi - 48708 Pi + 252192 Pi - 728704 Pi + 1114112 Pi - 647168) ------------------------------------------------------------------------- 6 2 2 (3 Pi - 8) i 3/2 + Pi ( 5 4 3 2 81405 Pi - 1064232 Pi + 3756672 Pi - 279552 Pi - 16543744 Pi + 17006592 1/2 / 2 7 7 6 ) (1/i) / (32 i (3 Pi - 8) ) + Pi (148716 Pi - 2387475 Pi / 5 4 3 2 + 17375904 Pi - 75938496 Pi + 212135936 Pi - 362123264 Pi / 8 3 3/2 7 + 333053952 Pi - 123994112) / (8 (3 Pi - 8) i ) - Pi (2770929 Pi / 6 5 4 3 - 62015544 Pi + 353236032 Pi - 56606208 Pi - 5454450688 Pi 2 1/2 / 3 + 18329206784 Pi - 23487315968 Pi + 10676600832) (1/i) / (256 i / 9 9 8 7 (3 Pi - 8) ) - Pi (10707552 Pi - 216147771 Pi + 2008054800 Pi 6 5 4 3 - 11539088064 Pi + 45346074624 Pi - 123578707968 Pi + 225576288256 Pi 2 / 10 - 257930035200 Pi + 164542545920 Pi - 44258295808) / (64 (3 Pi - 8) / 4 3/2 9 8 7 i ) - Pi (4074381 Pi + 9193045752 Pi - 92475737856 Pi 6 5 4 - 20431816704 Pi + 4062209728512 Pi - 23412903706624 Pi 3 2 + 63055904899072 Pi - 91076299849728 Pi + 67921770446848 Pi 1/2 / 4 11 11 - 20513971765248) (1/i) / (8192 i (3 Pi - 8) ) + Pi (770943744 Pi / 10 9 8 - 18683005185 Pi + 209790127008 Pi - 1475593495488 Pi 7 6 5 + 7321901617152 Pi - 26729264160768 Pi + 71519990972416 Pi 4 3 2 - 136347242725376 Pi + 177400721702912 Pi - 147838495358976 Pi / 12 5 3/2 + 70436926783488 Pi - 14468671078400) / (512 (3 Pi - 8) i ) + Pi ( / 11 10 9 58211389935 Pi - 356115386088 Pi - 1689364631232 Pi 8 7 6 - 15117157744128 Pi + 554008944992256 Pi - 4533219999940608 Pi 5 4 3 + 19102667798740992 Pi - 48122323093422080 Pi + 75189159299383296 Pi 2 - 71376966409781248 Pi + 37643413738225664 Pi - 8433554832752640) 1/2 / 5 13 1 (1/i) / (65536 i (3 Pi - 8) ) + O(----), / 6 i 2 1/2 1/2 2 3/2 1/2 (102 Pi - 160 Pi - 512) 2 225 2 (Pi - 2 Pi - 32/9) Pi (1/i) ----------------------------- + ------------------------------------------- 5/2 7/2 (3 Pi - 8) (3 Pi - 8) 1/2 3 2 5 2 Pi (1233 Pi - 7860 Pi + 20672 Pi - 25600) 1/2 3/2 + -------------------------------------------------- + 5 2 Pi 9/2 4 (3 Pi - 8) i 4 3 2 1/2 / (1053 Pi - 12060 Pi + 52740 Pi - 85504 Pi + 19456) (1/i) / (4 i / 11/2 1/2 5 4 3 (3 Pi - 8) ) - 2 Pi (706077 Pi - 8338968 Pi + 43005552 Pi 2 / 13/2 2 - 124475648 Pi + 189345792 Pi - 103546880) / (64 (3 Pi - 8) i ) - / 1/2 3/2 6 5 4 3 2 Pi (393660 Pi - 5762745 Pi + 35238510 Pi - 109804932 Pi 2 1/2 / 2 + 169707840 Pi - 105776128 Pi + 14499840) (1/i) / (16 i / 15/2 1/2 7 6 5 (3 Pi - 8) ) + 2 Pi (35807751 Pi - 663328116 Pi + 5463203760 Pi 4 3 2 - 26138053440 Pi + 76450974720 Pi - 130516074496 Pi + 114961678336 Pi / 17/2 3 1/2 3/2 8 - 39678115840) / (512 (3 Pi - 8) i ) + 5 2 Pi (1935495 Pi / 7 6 5 4 - 36328986 Pi + 295933905 Pi - 1341216144 Pi + 3592315044 Pi 3 2 1/2 - 5583447936 Pi + 4630702080 Pi - 1721925632 Pi + 194248704) (1/i) / 3 19/2 1/2 9 8 / (32 i (3 Pi - 8) ) - 2 Pi (5285220111 Pi - 164072729088 Pi / 7 6 5 + 2098199425248 Pi - 15102093171456 Pi + 67802060424960 Pi 4 3 2 - 195039354556416 Pi + 354358096232448 Pi - 387217650352128 Pi / 21/2 4 + 229795782721536 Pi - 56650618634240) / (16384 (3 Pi - 8) i ) - / 1/2 3/2 10 9 8 2 Pi (3389412600 Pi - 77919977835 Pi + 793931432220 Pi 7 6 5 - 4675451233608 Pi + 17350985580552 Pi - 41455246724688 Pi 4 3 2 + 62545499905792 Pi - 56144550780928 Pi + 26747604172800 Pi 1/2 / 4 23/2 - 5761112473600 Pi + 818644254720) (1/i) / (1024 i (3 Pi - 8) ) / 1/2 11 10 9 - 2 Pi (41613778917 Pi + 5893353558540 Pi - 154414484501472 Pi 8 7 6 + 1719468436544640 Pi - 11147840771086080 Pi + 46570836063992832 Pi 5 4 3 - 129997749305753600 Pi + 243353862481379328 Pi - 299238899035668480 Pi 2 / + 230042375584481280 Pi - 99643327166545920 Pi + 18519898980352000) / ( / 25/2 5 1/2 3/2 12 131072 (3 Pi - 8) i ) + 2 Pi (35154822150 Pi 11 10 9 - 957016886085 Pi + 11690777292660 Pi - 84152156117058 Pi 8 7 6 + 393060001756518 Pi - 1233161567599320 Pi + 2590623252744600 Pi 5 4 3 - 3490023579614720 Pi + 2656847039838208 Pi - 641649341104128 Pi 2 1/2 / - 466157733150720 Pi + 253101919436800 Pi + 13188905041920) (1/i) / / 3 2 5 27/2 1 105 Pi + 648 Pi - 2240 Pi - 2560 (1024 i (3 Pi - 8) ) + O(----), ---------------------------------- 6 3 i (3 Pi - 8) 2 3/2 1/2 (3870 Pi - 13392 Pi + 3840) Pi (1/i) + ------------------------------------------- 4 (3 Pi - 8) 4 3 2 Pi (14445 Pi - 69408 Pi + 49152 Pi + 245760 Pi - 512000) 3/2 + ----------------------------------------------------------- + Pi 5 2 (3 Pi - 8) i 4 3 2 1/2 (115425 Pi - 977496 Pi + 3449664 Pi - 5329408 Pi + 1761280) (1/i) / 6 6 5 4 / (4 i (3 Pi - 8) ) - Pi (26487 Pi + 228582 Pi - 4765104 Pi / 3 2 / 7 + 25011072 Pi - 66616320 Pi + 96288768 Pi - 51773440) / (4 (3 Pi - 8) / 2 3/2 6 5 4 3 i ) - Pi (24730353 Pi - 334472760 Pi + 1949564160 Pi - 6146632704 Pi 2 1/2 / 2 + 10436874240 Pi - 8288043008 Pi + 2288517120) (1/i) / (64 i / 8 8 7 6 (3 Pi - 8) ) - Pi (3726648 Pi - 70454205 Pi + 587276568 Pi 5 4 3 2 - 2789375808 Pi + 8378279424 Pi - 16791842816 Pi + 22055124992 Pi / 9 3 3/2 - 16446652416 Pi + 4959764480) / (8 (3 Pi - 8) i ) + Pi ( / 8 7 6 5 1678743387 Pi - 32167443816 Pi + 274058199360 Pi - 1342461067776 Pi 4 3 2 + 4040077307904 Pi - 7425853980672 Pi + 7942344278016 Pi 1/2 / 3 10 - 4524301352960 Pi + 1102766407680) (1/i) / (512 i (3 Pi - 8) ) + / 10 9 8 Pi (1141928928 Pi - 25309982601 Pi + 252017537544 Pi 7 6 5 - 1481228817984 Pi + 5717691339264 Pi - 15339531005952 Pi 4 3 2 + 29364732002304 Pi - 39565260488704 Pi + 34954631184384 Pi / 11 4 3/2 - 17468907061248 Pi + 3540663664640) / (128 (3 Pi - 8) i ) - Pi ( / 10 9 8 311822987067 Pi - 8488109429640 Pi + 102765010324608 Pi 7 6 5 - 724195956344832 Pi + 3250012663652352 Pi - 9571147284676608 Pi 4 3 2 + 18512640275906560 Pi - 23118560905658368 Pi + 18058583438524416 Pi 1/2 / 4 - 8311667016335360 Pi + 1850690670428160) (1/i) / (16384 i / 12 12 11 (3 Pi - 8) ) - Pi (62899939584 Pi - 1656426545721 Pi 10 9 8 + 19825855067448 Pi - 142629733306944 Pi + 689167648562688 Pi 7 6 5 - 2374103079936000 Pi + 6041739303124992 Pi - 11532525733675008 Pi 4 3 2 + 16332875297193984 Pi - 16424390358990848 Pi + 10780889617268736 Pi / 13 5 3/2 - 3970981806735360 Pi + 578746843136000) / (512 (3 Pi - 8) i ) + Pi / 12 11 10 (768806111151 Pi - 225236225786472 Pi + 5538799723414464 Pi 9 8 7 - 62934933437425152 Pi + 422282963024216064 Pi - 1834385139081216000 Pi 6 5 + 5365810145432961024 Pi - 10716449237064744960 Pi 4 3 + 14605895890691424256 Pi - 13480645987034726400 Pi 2 + 8333341377668579328 Pi - 3333216976171433984 Pi + 701695264146063360) 1/2 / 5 14 1 (1/i) / (131072 i (3 Pi - 8) ) + O(----), / 6 i 3 2 1/2 (1086 Pi + 336 Pi - 9856 Pi - 6144) 2 ------------------------------------------ 7/2 (3 Pi - 8) 1/2 3/2 3 2 1/2 7 2 Pi (945 Pi + 1812 Pi - 19488 Pi + 14080) (1/i) + ------------------------------------------------------------- 9/2 2 (3 Pi - 8) 1/2 4 3 2 7 2 Pi (44145 Pi - 259956 Pi + 463392 Pi - 96000 Pi - 512000) + ------------------------------------------------------------------- + 7 11/2 4 (3 Pi - 8) i 1/2 3/2 2 Pi 5 4 3 2 (62775 Pi - 484785 Pi + 1283004 Pi - 694176 Pi - 2002176 Pi + 1372160) 1/2 / 13/2 1/2 6 5 (1/i) / (4 i (3 Pi - 8) ) + 7 2 Pi (1240677 Pi - 12253248 Pi / 4 3 2 + 63367344 Pi - 209569152 Pi + 489389056 Pi - 751812608 Pi + 414187520) / 15/2 2 1/2 3/2 7 6 / (64 (3 Pi - 8) i ) - 7 2 Pi (1652643 Pi - 17385759 Pi / 5 4 3 2 + 55809945 Pi + 30113748 Pi - 621158304 Pi + 1521979136 Pi 1/2 / 2 17/2 1/2 - 1453064192 Pi + 514129920) (1/i) / (16 i (3 Pi - 8) ) - 2 Pi / 8 7 6 5 (3193923231 Pi - 49253124708 Pi + 338758895952 Pi - 1361631044160 Pi 4 3 2 + 3519346143744 Pi - 6082599137280 Pi + 6878711971840 Pi / 19/2 3 1/2 - 4379532853248 Pi + 1110987243520) / (512 (3 Pi - 8) i ) - 2 / 3/2 9 8 7 6 Pi (149722020 Pi - 6421058973 Pi + 92979796644 Pi - 681505475364 Pi 5 4 3 + 2937288804336 Pi - 7882700499072 Pi + 13201190489088 Pi 2 1/2 / - 13266408718336 Pi + 7398504529920 Pi - 1821061939200) (1/i) / (128 / 3 21/2 1/2 10 9 i (3 Pi - 8) ) + 2 Pi (1247160524877 Pi - 25787484797688 Pi 8 7 6 + 242725943192544 Pi - 1373318383127040 Pi + 5169403997662464 Pi 5 4 3 - 13494167507552256 Pi + 24605135818604544 Pi - 30496446803607552 Pi 2 / + 23827288751603712 Pi - 10072570187481088 Pi + 1586217321758720) / ( / 23/2 4 1/2 3/2 11 16384 (3 Pi - 8) i ) + 7 2 Pi (8188324830 Pi 10 9 8 - 213763514535 Pi + 2551330816596 Pi - 18195198428484 Pi 7 6 5 + 85550706792054 Pi - 277011418273224 Pi + 627092060811072 Pi 4 3 2 - 986962104100352 Pi + 1054456168120320 Pi - 730950315278336 Pi 1/2 / 4 25/2 + 299875434496000 Pi - 56196794941440) (1/i) / (512 i (3 Pi - 8) / 1/2 12 11 ) - 7 2 Pi (11349920778345 Pi - 304686201553956 Pi 10 9 8 + 3790377978377472 Pi - 28899484621928064 Pi + 149866402681923840 Pi 7 6 - 553657085886379008 Pi + 1483753341572161536 Pi 5 4 - 2881343897249185792 Pi + 3975177636777295872 Pi 3 2 - 3730624551793459200 Pi + 2196758744782602240 Pi / 27/2 5 - 689774152919285760 Pi + 74079595921408000) / (131072 (3 Pi - 8) i / 1/2 3/2 13 12 ) - 7 2 Pi (2516862060720 Pi - 74466229597713 Pi 11 10 9 + 1013792528479644 Pi - 8378066772395352 Pi + 46741562444142576 Pi 8 7 - 185414166664363152 Pi + 536770563145509312 Pi 6 5 - 1145649369688201728 Pi + 1798958742271143936 Pi 4 3 - 2050351401829203968 Pi + 1650685889382187008 Pi 2 - 892319885076463616 Pi + 291508069437800448 Pi - 43433909047787520) 1/2 / 5 29/2 1 (1/i) / (8192 i (3 Pi - 8) ) + O(----), / 6 i 4 3 2 945 Pi + 11232 Pi - 18816 Pi - 71680 Pi - 28672 -------------------------------------------------- 4 (3 Pi - 8) 3 2 3/2 1/2 (81144 Pi - 187008 Pi - 387072 Pi + 544768) Pi (1/i) + ------------------------------------------------------------ + Pi ( 5 (3 Pi - 8) 5 4 3 2 176715 Pi - 211464 Pi - 3444864 Pi + 10192896 Pi - 6852608 Pi - 5734400 / 6 3/2 5 4 3 ) / ((3 Pi - 8) i) + Pi (2962197 Pi - 24505200 Pi + 75580224 Pi / 2 1/2 / 7 - 95516160 Pi + 18751488 Pi + 20873216) (1/i) / (i (3 Pi - 8) ) + Pi / 7 6 5 4 (12646935 Pi - 139043088 Pi + 637655616 Pi - 1451911680 Pi 3 2 / + 1444220928 Pi + 203227136 Pi - 1965293568 Pi + 1159725056) / (4 / 8 2 3/2 7 6 5 (3 Pi - 8) i ) - Pi (134972649 Pi - 1887976080 Pi + 10095463296 Pi 4 3 2 - 25104327168 Pi + 20244123648 Pi + 31116623872 Pi - 65142784000 Pi 1/2 / 2 9 9 + 33250344960) (1/i) / (16 i (3 Pi - 8) ) - Pi (336367161 Pi / 8 7 6 5 - 4839849954 Pi + 28174033440 Pi - 79272428544 Pi + 76743038976 Pi 4 3 2 + 160975896576 Pi - 606755487744 Pi + 832611287040 Pi - 530537512960 Pi / 10 3 3/2 9 + 111098724352) / (8 (3 Pi - 8) i ) - Pi (22112676081 Pi / 8 7 6 - 378181821456 Pi + 2953158816960 Pi - 13944141762048 Pi 5 4 3 + 44202477981696 Pi - 97400359452672 Pi + 145835321196544 Pi 2 1/2 / - 137422857830400 Pi + 72593033920512 Pi - 17134100938752) (1/i) / ( / 3 11 11 10 128 i (3 Pi - 8) ) + Pi (7619604228 Pi - 122593057011 Pi 9 8 7 + 666957698784 Pi - 183572179200 Pi - 15534958934016 Pi 6 5 4 + 89163394744320 Pi - 265005494304768 Pi + 491414964666368 Pi 3 2 - 588348284469248 Pi + 432369844617216 Pi - 164672267354112 Pi / 12 4 3/2 11 + 19827716521984) / (32 (3 Pi - 8) i ) + Pi (14063594651973 Pi / 10 9 8 - 314735922228528 Pi + 3230164053536256 Pi - 20157910706741760 Pi 7 6 5 + 85149698336317440 Pi - 254987966548279296 Pi + 547392212815577088 Pi 4 3 2 - 828687819932172288 Pi + 848441544573714432 Pi - 547714863950987264 Pi 1/2 / 4 + 197576625989091328 Pi - 29917994859626496) (1/i) / (4096 i / 13 13 12 (3 Pi - 8) ) + Pi (379701131328 Pi - 13578413819553 Pi 11 10 9 + 218752219226448 Pi - 2068657320726144 Pi + 12776076005446656 Pi 8 7 6 - 54547132817977344 Pi + 166420162034663424 Pi - 369700861343956992 Pi 5 4 3 + 601845811675922432 Pi - 710220601607847936 Pi + 583330273588936704 Pi 2 / - 304093960266055680 Pi + 82515890943295488 Pi - 6481964643123200) / ( / 14 5 3/2 13 256 (3 Pi - 8) i ) - Pi (1222412422254741 Pi 12 11 - 33923305359382320 Pi + 438407240333262912 Pi 10 9 - 3505918772208416256 Pi + 19363014219956797440 Pi 8 7 - 77713205767036305408 Pi + 231422659053979435008 Pi 6 5 - 512105291003266596864 Pi + 831100185203852705792 Pi 4 3 - 963268026986135552000 Pi + 763770745055424282624 Pi 2 - 385723890689881669632 Pi + 108015592541106733056 Pi 1/2 / 5 15 1 - 11637219523509092352) (1/i) / (32768 i (3 Pi - 8) ) + O(----), / 6 i 4 3 2 1/2 (13170 Pi + 30528 Pi - 145152 Pi - 233472 Pi - 65536) 2 ------------------------------------------------------------- + 9/2 (3 Pi - 8) 4 3 2 1/2 3/2 (51030 Pi + 354402 Pi - 1805760 Pi - 145152 Pi + 2322432) 2 Pi 1/2 / 11/2 1/2 5 4 (1/i) / (3 Pi - 8) + 3 2 Pi (3400677 Pi - 16424244 Pi / 3 2 / + 9586176 Pi + 47427072 Pi - 58318848 Pi - 22937600) / (4 / 13/2 1/2 3/2 6 5 4 (3 Pi - 8) i) + 3 2 Pi (2934225 Pi - 12170331 Pi - 31198926 Pi 3 2 1/2 / + 251031456 Pi - 464124672 Pi + 218652672 Pi + 52756480) (1/i) / (2 / 15/2 1/2 7 6 i (3 Pi - 8) ) + 3 2 Pi (1186330131 Pi - 12692100600 Pi 5 4 3 2 + 55687961232 Pi - 124240421376 Pi + 138354739200 Pi - 54872014848 Pi / 17/2 2 1/2 3/2 - 27800371200 Pi + 18555600896) / (64 (3 Pi - 8) i ) + 3 2 Pi / 8 7 6 5 (206263260 Pi - 3172233537 Pi + 22143472848 Pi - 86633783532 Pi 4 3 2 + 197223886080 Pi - 244393216512 Pi + 115671441408 Pi + 36684562432 Pi 1/2 / 2 19/2 1/2 - 43467669504) (1/i) / (16 i (3 Pi - 8) ) - 2 Pi ( / 9 8 7 221558382027 Pi - 3523366329588 Pi + 23696523492912 Pi 6 5 4 - 86989177348416 Pi + 183241875124224 Pi - 200894671429632 Pi 3 2 + 44534791798784 Pi + 138610533728256 Pi - 124062277828608 Pi / 21/2 3 1/2 3/2 + 21330955075584) / (512 (3 Pi - 8) i ) - 9 2 Pi ( / 10 9 8 7 7049050920 Pi - 123892410573 Pi + 946496444328 Pi - 4038747981444 Pi 6 5 4 + 10139901180072 Pi - 13672001006208 Pi + 3743987051520 Pi 3 2 + 17556683587584 Pi - 27616953237504 Pi + 16907088429056 Pi 1/2 / 3 23/2 1/2 - 3768900911104) (1/i) / (64 i (3 Pi - 8) ) - 2 Pi ( / 11 10 9 20430982218861 Pi - 335296640001456 Pi + 2615740610964768 Pi 8 7 6 - 13654533537144576 Pi + 56390194704708864 Pi - 191640363278819328 Pi 5 4 + 504217306866548736 Pi - 948083530438017024 Pi 3 2 + 1184506163027247104 Pi - 889066357423865856 Pi + 328890316146868224 Pi / 25/2 4 1/2 3/2 - 30455372577767424) / (16384 (3 Pi - 8) i ) + 2 Pi ( / 12 11 10 10659388486320 Pi - 226059952467105 Pi + 2051707730023620 Pi 9 8 7 - 10092319189010136 Pi + 26500347481873392 Pi - 17275759082116944 Pi 6 5 4 - 126517223075544576 Pi + 501604210195494912 Pi - 951735959264919552 Pi 3 2 + 1070757972046512128 Pi - 710690882206040064 Pi + 245629609020751872 Pi 1/2 / 4 27/2 1/2 - 29771293809180672) (1/i) / (1024 i (3 Pi - 8) ) + 2 Pi ( / 13 12 11 9615618398620737 Pi - 231897873554319636 Pi + 2577233980700948448 Pi 10 9 - 17555819629741188480 Pi + 82188363072956656896 Pi 8 7 - 280402972553106662400 Pi + 717993269025205911552 Pi 6 5 - 1391324026121373745152 Pi + 2020260364203646255104 Pi 4 3 - 2137438038145628110848 Pi + 1564626794235748679680 Pi 2 - 721033767056761159680 Pi + 170429881227279335424 Pi / 29/2 5 1/2 3/2 - 9956297691837235200) / (131072 (3 Pi - 8) i ) - 3 2 Pi ( / 14 13 12 62950121786160 Pi - 1319513707621647 Pi + 8837999116429068 Pi 11 10 + 14827686138188928 Pi - 641310998346130032 Pi 9 8 + 5070901516480793232 Pi - 23177537154844970976 Pi 7 6 + 71260902360471654912 Pi - 154964894249700519936 Pi 5 4 + 241660923510239133696 Pi - 267415966497712898048 Pi 3 2 + 202273799806283415552 Pi - 96493929729292763136 Pi 1/2 / 5 + 24343542727245299712 Pi - 1936385318206832640) (1/i) / (4096 i / 31/2 1 (3 Pi - 8) ) + O(----), 6 i 5 4 3 2 3 (3465 Pi + 62200 Pi - 21120 Pi - 494592 Pi - 471040 Pi - 98304) --------------------------------------------------------------------- + 5 (3 Pi - 8) 4 3 2 3/2 (1626750 Pi - 1106160 Pi - 18051840 Pi + 11704320 Pi + 17203200) Pi 1/2 / 6 6 5 4 (1/i) / (3 Pi - 8) + 15 Pi (274995 Pi + 916344 Pi - 12013632 Pi / 3 2 / 7 + 21467136 Pi + 5419008 Pi - 27721728 Pi - 6553600) / ((3 Pi - 8) i) / 3/2 6 5 4 3 + 15 Pi (39874923 Pi - 292323096 Pi + 650382272 Pi - 30887424 Pi 2 1/2 / - 1542610944 Pi + 1211170816 Pi + 145752064) (1/i) / (4 i / 8 8 7 6 (3 Pi - 8) ) + 3 Pi (284143545 Pi - 1997737830 Pi + 1654218000 Pi 5 4 3 2 + 23292092032 Pi - 88670665728 Pi + 126643126272 Pi - 65964474368 Pi / 9 2 3/2 - 8124366848 Pi + 6627000320) / (4 (3 Pi - 8) i ) + 3 Pi ( / 8 7 6 5 42264235485 Pi - 556699667400 Pi + 3186362350080 Pi - 10172821954560 Pi 4 3 2 + 19316946309120 Pi - 21253706776576 Pi + 11232857292800 Pi 1/2 / 2 10 - 688968957952 Pi - 1227085578240) (1/i) / (64 i (3 Pi - 8) ) + Pi / 10 9 8 7 (3261287205 Pi - 104752787490 Pi + 1298177753040 Pi - 8254390499712 Pi 6 5 4 + 30281809185792 Pi - 66358159073280 Pi + 83677588684800 Pi 3 2 - 50223267184640 Pi - 986047315968 Pi + 13453951500288 Pi / 11 3 3/2 10 - 1904549560320) / (8 (3 Pi - 8) i ) - 5 Pi (2962656131949 Pi / 9 8 7 - 53713233641736 Pi + 426761792268480 Pi - 1942460404379136 Pi 6 5 4 + 5519139155546112 Pi - 9901116793847808 Pi + 10525669198135296 Pi 3 2 - 5029782691512320 Pi - 981955856302080 Pi + 1841520646815744 Pi 1/2 / 3 12 - 385271451353088) (1/i) / (512 i (3 Pi - 8) ) - Pi ( / 12 11 10 5366900466936 Pi - 111356293021083 Pi + 1046934575794728 Pi 9 8 7 - 5825153277893952 Pi + 20984631631308288 Pi - 50183619675254784 Pi 6 5 4 + 77679401157033984 Pi - 68524911458254848 Pi + 12572235569037312 Pi 3 2 + 43253140283195392 Pi - 48384368078487552 Pi + 18806003931807744 Pi / 13 4 3/2 - 1359614847221760) / (128 (3 Pi - 8) i ) + Pi ( / 12 11 10 1465626345516261 Pi - 37679884502621544 Pi + 419190597331560576 Pi 9 8 - 2673098265793987584 Pi + 10727632849620934656 Pi 7 6 - 27414206448491495424 Pi + 40803300727257563136 Pi 5 4 - 18698856772134764544 Pi - 50250141231537455104 Pi 3 2 + 108437521231643672576 Pi - 92446164193483161600 Pi 1/2 / 4 + 34828862613356544000 Pi - 3392374077590077440) (1/i) / (16384 i / 14 14 13 (3 Pi - 8) ) + Pi (10605311884512 Pi - 260400586731615 Pi 12 11 10 + 2860256763844920 Pi - 18377323159706604 Pi + 75332353862112480 Pi 9 8 7 - 198451751086259712 Pi + 301967428366970880 Pi - 92015299292774400 Pi 6 5 - 697903583741214720 Pi + 1763963105186938880 Pi 4 3 - 2249598040820154368 Pi + 1718707819265392640 Pi 2 / - 755490470410321920 Pi + 156084029274193920 Pi - 6944962117632000) / ( / 15 5 3/2 14 16 (3 Pi - 8) i ) + Pi (363073114458974907 Pi 13 12 - 9292478157285751512 Pi + 109975663793974691520 Pi 11 10 - 802009043877679879680 Pi + 4062572219921041981440 Pi 9 8 - 15307675778431331205120 Pi + 44782669423650173091840 Pi 7 6 - 103877301379203826974720 Pi + 190553427164491646238720 Pi 5 4 - 269578352459267985899520 Pi + 281574482435066537443328 Pi 3 2 - 204098935423993588482048 Pi + 93247688953331491799040 Pi 1/2 / - 22024236420456995880960 Pi + 1326510553220587192320) (1/i) / ( / 5 16 1 131072 i (3 Pi - 8) ) + O(----)] 6 i and in Maple notation [0, 1, (10*Pi-32)*2^(1/2)/(3*Pi-8)^(3/2)+3/2*2^(1/2)*(9*Pi-28)*(1/i)^(1/2)*Pi^( 3/2)/(3*Pi-8)^(5/2)-1/4*2^(1/2)*Pi*(63*Pi^2-708*Pi+1600)/(3*Pi-8)^(7/2)/i-1/4*2 ^(1/2)*(243*Pi^3-1485*Pi^2+1676*Pi+1856)/i*(1/i)^(1/2)*Pi^(3/2)/(3*Pi-8)^(9/2)+ 1/64*2^(1/2)*Pi*(9801*Pi^4-176904*Pi^3+961776*Pi^2-1991424*Pi+1294336)/(3*Pi-8) ^(11/2)/i^2+1/16*2^(1/2)*(6561*Pi^5-56619*Pi^4+81261*Pi^3+480756*Pi^2-1591936* Pi+1244160)/i^2*(1/i)^(1/2)*Pi^(3/2)/(3*Pi-8)^(13/2)-1/512*2^(1/2)*Pi*(741393* Pi^6-18719748*Pi^5+161472528*Pi^4-655116096*Pi^3+1345827840*Pi^2-1330249728*Pi+ 495976448)/(3*Pi-8)^(15/2)/i^3-1/128*2^(1/2)*(393660*Pi^7-4223097*Pi^6+3911652* Pi^5+126111276*Pi^4-707568336*Pi^3+1633140480*Pi^2-1752485888*Pi+715849728)/i^3 *(1/i)^(1/2)*Pi^(3/2)/(3*Pi-8)^(17/2)+1/16384*2^(1/2)*Pi*(218028591*Pi^8-\ 7190856000*Pi^7+85562508384*Pi^6-518519847168*Pi^5+1798599700224*Pi^4-\ 3693560082432*Pi^3+4397326663680*Pi^2-2774481764352*Pi+708132732928)/(3*Pi-8)^( 19/2)/i^4+1/512*2^(1/2)*(12400290*Pi^9-155082357*Pi^8-61583004*Pi^7+11428086852 *Pi^6-87461955630*Pi^5+327527209928*Pi^4-703565263360*Pi^3+878291914752*Pi^2-\ 591057649664*Pi+165381931008)/i^4*(1/i)^(1/2)*Pi^(3/2)/(3*Pi-8)^(21/2)-1/131072 *2^(1/2)*Pi*(15653240361*Pi^10-647718165036*Pi^9+9904673926944*Pi^8-\ 80251272360576*Pi^7+393517933012224*Pi^6-1236454635568128*Pi^5+2534737216389120 *Pi^4-3351731383566336*Pi^3+2731484343435264*Pi^2-1232740439556096*Pi+ 231498737254400)/(3*Pi-8)^(23/2)/i^5-1/8192*2^(1/2)*(1607077584*Pi^11-\ 22608385875*Pi^10-56053378620*Pi^9+3336117562008*Pi^8-32132432146608*Pi^7+ 165281284460880*Pi^6-530461778010816*Pi^5+1110874084613120*Pi^4-\ 1517401615269888*Pi^3+1300815649701888*Pi^2-633420363333632*Pi+133093184765952) /i^5*(1/i)^(1/2)*Pi^(3/2)/(3*Pi-8)^(25/2)+O(1/i^6), (15*Pi^2+16*Pi-192)/(3*Pi-8 )^2+(132*Pi-416)*Pi^(3/2)*(1/i)^(1/2)/(3*Pi-8)^3+1/2*Pi*(459*Pi^3-3384*Pi^2+ 10176*Pi-12800)/(3*Pi-8)^4/i-1/2*Pi^(3/2)*(999*Pi^3-7656*Pi^2+13376*Pi+2560)/i* (1/i)^(1/2)/(3*Pi-8)^5-1/2*Pi*(4131*Pi^5-48708*Pi^4+252192*Pi^3-728704*Pi^2+ 1114112*Pi-647168)/(3*Pi-8)^6/i^2+1/32*Pi^(3/2)*(81405*Pi^5-1064232*Pi^4+ 3756672*Pi^3-279552*Pi^2-16543744*Pi+17006592)/i^2*(1/i)^(1/2)/(3*Pi-8)^7+1/8* Pi*(148716*Pi^7-2387475*Pi^6+17375904*Pi^5-75938496*Pi^4+212135936*Pi^3-\ 362123264*Pi^2+333053952*Pi-123994112)/(3*Pi-8)^8/i^3-1/256*Pi^(3/2)*(2770929* Pi^7-62015544*Pi^6+353236032*Pi^5-56606208*Pi^4-5454450688*Pi^3+18329206784*Pi^ 2-23487315968*Pi+10676600832)/i^3*(1/i)^(1/2)/(3*Pi-8)^9-1/64*Pi*(10707552*Pi^9 -216147771*Pi^8+2008054800*Pi^7-11539088064*Pi^6+45346074624*Pi^5-123578707968* Pi^4+225576288256*Pi^3-257930035200*Pi^2+164542545920*Pi-44258295808)/(3*Pi-8)^ 10/i^4-1/8192*Pi^(3/2)*(4074381*Pi^9+9193045752*Pi^8-92475737856*Pi^7-\ 20431816704*Pi^6+4062209728512*Pi^5-23412903706624*Pi^4+63055904899072*Pi^3-\ 91076299849728*Pi^2+67921770446848*Pi-20513971765248)/i^4*(1/i)^(1/2)/(3*Pi-8)^ 11+1/512*Pi*(770943744*Pi^11-18683005185*Pi^10+209790127008*Pi^9-1475593495488* Pi^8+7321901617152*Pi^7-26729264160768*Pi^6+71519990972416*Pi^5-136347242725376 *Pi^4+177400721702912*Pi^3-147838495358976*Pi^2+70436926783488*Pi-\ 14468671078400)/(3*Pi-8)^12/i^5+1/65536*Pi^(3/2)*(58211389935*Pi^11-\ 356115386088*Pi^10-1689364631232*Pi^9-15117157744128*Pi^8+554008944992256*Pi^7-\ 4533219999940608*Pi^6+19102667798740992*Pi^5-48122323093422080*Pi^4+ 75189159299383296*Pi^3-71376966409781248*Pi^2+37643413738225664*Pi-\ 8433554832752640)/i^5*(1/i)^(1/2)/(3*Pi-8)^13+O(1/i^6), (102*Pi^2-160*Pi-512)*2 ^(1/2)/(3*Pi-8)^(5/2)+225*2^(1/2)*(Pi^2-2*Pi-32/9)*Pi^(3/2)*(1/i)^(1/2)/(3*Pi-8 )^(7/2)+5/4*2^(1/2)*Pi*(1233*Pi^3-7860*Pi^2+20672*Pi-25600)/(3*Pi-8)^(9/2)/i+5/ 4*2^(1/2)*Pi^(3/2)*(1053*Pi^4-12060*Pi^3+52740*Pi^2-85504*Pi+19456)/i*(1/i)^(1/ 2)/(3*Pi-8)^(11/2)-1/64*2^(1/2)*Pi*(706077*Pi^5-8338968*Pi^4+43005552*Pi^3-\ 124475648*Pi^2+189345792*Pi-103546880)/(3*Pi-8)^(13/2)/i^2-1/16*2^(1/2)*Pi^(3/2 )*(393660*Pi^6-5762745*Pi^5+35238510*Pi^4-109804932*Pi^3+169707840*Pi^2-\ 105776128*Pi+14499840)/i^2*(1/i)^(1/2)/(3*Pi-8)^(15/2)+1/512*2^(1/2)*Pi*( 35807751*Pi^7-663328116*Pi^6+5463203760*Pi^5-26138053440*Pi^4+76450974720*Pi^3-\ 130516074496*Pi^2+114961678336*Pi-39678115840)/(3*Pi-8)^(17/2)/i^3+5/32*2^(1/2) *Pi^(3/2)*(1935495*Pi^8-36328986*Pi^7+295933905*Pi^6-1341216144*Pi^5+3592315044 *Pi^4-5583447936*Pi^3+4630702080*Pi^2-1721925632*Pi+194248704)/i^3*(1/i)^(1/2)/ (3*Pi-8)^(19/2)-1/16384*2^(1/2)*Pi*(5285220111*Pi^9-164072729088*Pi^8+ 2098199425248*Pi^7-15102093171456*Pi^6+67802060424960*Pi^5-195039354556416*Pi^4 +354358096232448*Pi^3-387217650352128*Pi^2+229795782721536*Pi-56650618634240)/( 3*Pi-8)^(21/2)/i^4-1/1024*2^(1/2)*Pi^(3/2)*(3389412600*Pi^10-77919977835*Pi^9+ 793931432220*Pi^8-4675451233608*Pi^7+17350985580552*Pi^6-41455246724688*Pi^5+ 62545499905792*Pi^4-56144550780928*Pi^3+26747604172800*Pi^2-5761112473600*Pi+ 818644254720)/i^4*(1/i)^(1/2)/(3*Pi-8)^(23/2)-1/131072*2^(1/2)*Pi*(41613778917* Pi^11+5893353558540*Pi^10-154414484501472*Pi^9+1719468436544640*Pi^8-\ 11147840771086080*Pi^7+46570836063992832*Pi^6-129997749305753600*Pi^5+ 243353862481379328*Pi^4-299238899035668480*Pi^3+230042375584481280*Pi^2-\ 99643327166545920*Pi+18519898980352000)/(3*Pi-8)^(25/2)/i^5+1/1024*2^(1/2)*Pi^( 3/2)*(35154822150*Pi^12-957016886085*Pi^11+11690777292660*Pi^10-84152156117058* Pi^9+393060001756518*Pi^8-1233161567599320*Pi^7+2590623252744600*Pi^6-\ 3490023579614720*Pi^5+2656847039838208*Pi^4-641649341104128*Pi^3-\ 466157733150720*Pi^2+253101919436800*Pi+13188905041920)/i^5*(1/i)^(1/2)/(3*Pi-8 )^(27/2)+O(1/i^6), (105*Pi^3+648*Pi^2-2240*Pi-2560)/(3*Pi-8)^3+(3870*Pi^2-13392 *Pi+3840)*Pi^(3/2)*(1/i)^(1/2)/(3*Pi-8)^4+1/2*Pi*(14445*Pi^4-69408*Pi^3+49152* Pi^2+245760*Pi-512000)/(3*Pi-8)^5/i+1/4*Pi^(3/2)*(115425*Pi^4-977496*Pi^3+ 3449664*Pi^2-5329408*Pi+1761280)/i*(1/i)^(1/2)/(3*Pi-8)^6-1/4*Pi*(26487*Pi^6+ 228582*Pi^5-4765104*Pi^4+25011072*Pi^3-66616320*Pi^2+96288768*Pi-51773440)/(3* Pi-8)^7/i^2-1/64*Pi^(3/2)*(24730353*Pi^6-334472760*Pi^5+1949564160*Pi^4-\ 6146632704*Pi^3+10436874240*Pi^2-8288043008*Pi+2288517120)/i^2*(1/i)^(1/2)/(3* Pi-8)^8-1/8*Pi*(3726648*Pi^8-70454205*Pi^7+587276568*Pi^6-2789375808*Pi^5+ 8378279424*Pi^4-16791842816*Pi^3+22055124992*Pi^2-16446652416*Pi+4959764480)/(3 *Pi-8)^9/i^3+1/512*Pi^(3/2)*(1678743387*Pi^8-32167443816*Pi^7+274058199360*Pi^6 -1342461067776*Pi^5+4040077307904*Pi^4-7425853980672*Pi^3+7942344278016*Pi^2-\ 4524301352960*Pi+1102766407680)/i^3*(1/i)^(1/2)/(3*Pi-8)^10+1/128*Pi*( 1141928928*Pi^10-25309982601*Pi^9+252017537544*Pi^8-1481228817984*Pi^7+ 5717691339264*Pi^6-15339531005952*Pi^5+29364732002304*Pi^4-39565260488704*Pi^3+ 34954631184384*Pi^2-17468907061248*Pi+3540663664640)/(3*Pi-8)^11/i^4-1/16384*Pi ^(3/2)*(311822987067*Pi^10-8488109429640*Pi^9+102765010324608*Pi^8-\ 724195956344832*Pi^7+3250012663652352*Pi^6-9571147284676608*Pi^5+ 18512640275906560*Pi^4-23118560905658368*Pi^3+18058583438524416*Pi^2-\ 8311667016335360*Pi+1850690670428160)/i^4*(1/i)^(1/2)/(3*Pi-8)^12-1/512*Pi*( 62899939584*Pi^12-1656426545721*Pi^11+19825855067448*Pi^10-142629733306944*Pi^9 +689167648562688*Pi^8-2374103079936000*Pi^7+6041739303124992*Pi^6-\ 11532525733675008*Pi^5+16332875297193984*Pi^4-16424390358990848*Pi^3+ 10780889617268736*Pi^2-3970981806735360*Pi+578746843136000)/(3*Pi-8)^13/i^5+1/ 131072*Pi^(3/2)*(768806111151*Pi^12-225236225786472*Pi^11+5538799723414464*Pi^ 10-62934933437425152*Pi^9+422282963024216064*Pi^8-1834385139081216000*Pi^7+ 5365810145432961024*Pi^6-10716449237064744960*Pi^5+14605895890691424256*Pi^4-\ 13480645987034726400*Pi^3+8333341377668579328*Pi^2-3333216976171433984*Pi+ 701695264146063360)/i^5*(1/i)^(1/2)/(3*Pi-8)^14+O(1/i^6), (1086*Pi^3+336*Pi^2-\ 9856*Pi-6144)*2^(1/2)/(3*Pi-8)^(7/2)+7/2*2^(1/2)*Pi^(3/2)*(945*Pi^3+1812*Pi^2-\ 19488*Pi+14080)*(1/i)^(1/2)/(3*Pi-8)^(9/2)+7/4*2^(1/2)*Pi*(44145*Pi^4-259956*Pi ^3+463392*Pi^2-96000*Pi-512000)/(3*Pi-8)^(11/2)/i+7/4*2^(1/2)*Pi^(3/2)*(62775* Pi^5-484785*Pi^4+1283004*Pi^3-694176*Pi^2-2002176*Pi+1372160)/i*(1/i)^(1/2)/(3* Pi-8)^(13/2)+7/64*2^(1/2)*Pi*(1240677*Pi^6-12253248*Pi^5+63367344*Pi^4-\ 209569152*Pi^3+489389056*Pi^2-751812608*Pi+414187520)/(3*Pi-8)^(15/2)/i^2-7/16* 2^(1/2)*Pi^(3/2)*(1652643*Pi^7-17385759*Pi^6+55809945*Pi^5+30113748*Pi^4-\ 621158304*Pi^3+1521979136*Pi^2-1453064192*Pi+514129920)/i^2*(1/i)^(1/2)/(3*Pi-8 )^(17/2)-1/512*2^(1/2)*Pi*(3193923231*Pi^8-49253124708*Pi^7+338758895952*Pi^6-\ 1361631044160*Pi^5+3519346143744*Pi^4-6082599137280*Pi^3+6878711971840*Pi^2-\ 4379532853248*Pi+1110987243520)/(3*Pi-8)^(19/2)/i^3-1/128*2^(1/2)*Pi^(3/2)*( 149722020*Pi^9-6421058973*Pi^8+92979796644*Pi^7-681505475364*Pi^6+2937288804336 *Pi^5-7882700499072*Pi^4+13201190489088*Pi^3-13266408718336*Pi^2+7398504529920* Pi-1821061939200)/i^3*(1/i)^(1/2)/(3*Pi-8)^(21/2)+1/16384*2^(1/2)*Pi*( 1247160524877*Pi^10-25787484797688*Pi^9+242725943192544*Pi^8-1373318383127040* Pi^7+5169403997662464*Pi^6-13494167507552256*Pi^5+24605135818604544*Pi^4-\ 30496446803607552*Pi^3+23827288751603712*Pi^2-10072570187481088*Pi+ 1586217321758720)/(3*Pi-8)^(23/2)/i^4+7/512*2^(1/2)*Pi^(3/2)*(8188324830*Pi^11-\ 213763514535*Pi^10+2551330816596*Pi^9-18195198428484*Pi^8+85550706792054*Pi^7-\ 277011418273224*Pi^6+627092060811072*Pi^5-986962104100352*Pi^4+1054456168120320 *Pi^3-730950315278336*Pi^2+299875434496000*Pi-56196794941440)/i^4*(1/i)^(1/2)/( 3*Pi-8)^(25/2)-7/131072*2^(1/2)*Pi*(11349920778345*Pi^12-304686201553956*Pi^11+ 3790377978377472*Pi^10-28899484621928064*Pi^9+149866402681923840*Pi^8-\ 553657085886379008*Pi^7+1483753341572161536*Pi^6-2881343897249185792*Pi^5+ 3975177636777295872*Pi^4-3730624551793459200*Pi^3+2196758744782602240*Pi^2-\ 689774152919285760*Pi+74079595921408000)/(3*Pi-8)^(27/2)/i^5-7/8192*2^(1/2)*Pi^ (3/2)*(2516862060720*Pi^13-74466229597713*Pi^12+1013792528479644*Pi^11-\ 8378066772395352*Pi^10+46741562444142576*Pi^9-185414166664363152*Pi^8+ 536770563145509312*Pi^7-1145649369688201728*Pi^6+1798958742271143936*Pi^5-\ 2050351401829203968*Pi^4+1650685889382187008*Pi^3-892319885076463616*Pi^2+ 291508069437800448*Pi-43433909047787520)/i^5*(1/i)^(1/2)/(3*Pi-8)^(29/2)+O(1/i^ 6), (945*Pi^4+11232*Pi^3-18816*Pi^2-71680*Pi-28672)/(3*Pi-8)^4+(81144*Pi^3-\ 187008*Pi^2-387072*Pi+544768)*Pi^(3/2)*(1/i)^(1/2)/(3*Pi-8)^5+Pi*(176715*Pi^5-\ 211464*Pi^4-3444864*Pi^3+10192896*Pi^2-6852608*Pi-5734400)/(3*Pi-8)^6/i+Pi^(3/2 )*(2962197*Pi^5-24505200*Pi^4+75580224*Pi^3-95516160*Pi^2+18751488*Pi+20873216) /i*(1/i)^(1/2)/(3*Pi-8)^7+1/4*Pi*(12646935*Pi^7-139043088*Pi^6+637655616*Pi^5-\ 1451911680*Pi^4+1444220928*Pi^3+203227136*Pi^2-1965293568*Pi+1159725056)/(3*Pi-\ 8)^8/i^2-1/16*Pi^(3/2)*(134972649*Pi^7-1887976080*Pi^6+10095463296*Pi^5-\ 25104327168*Pi^4+20244123648*Pi^3+31116623872*Pi^2-65142784000*Pi+33250344960)/ i^2*(1/i)^(1/2)/(3*Pi-8)^9-1/8*Pi*(336367161*Pi^9-4839849954*Pi^8+28174033440* Pi^7-79272428544*Pi^6+76743038976*Pi^5+160975896576*Pi^4-606755487744*Pi^3+ 832611287040*Pi^2-530537512960*Pi+111098724352)/(3*Pi-8)^10/i^3-1/128*Pi^(3/2)* (22112676081*Pi^9-378181821456*Pi^8+2953158816960*Pi^7-13944141762048*Pi^6+ 44202477981696*Pi^5-97400359452672*Pi^4+145835321196544*Pi^3-137422857830400*Pi ^2+72593033920512*Pi-17134100938752)/i^3*(1/i)^(1/2)/(3*Pi-8)^11+1/32*Pi*( 7619604228*Pi^11-122593057011*Pi^10+666957698784*Pi^9-183572179200*Pi^8-\ 15534958934016*Pi^7+89163394744320*Pi^6-265005494304768*Pi^5+491414964666368*Pi ^4-588348284469248*Pi^3+432369844617216*Pi^2-164672267354112*Pi+19827716521984) /(3*Pi-8)^12/i^4+1/4096*Pi^(3/2)*(14063594651973*Pi^11-314735922228528*Pi^10+ 3230164053536256*Pi^9-20157910706741760*Pi^8+85149698336317440*Pi^7-\ 254987966548279296*Pi^6+547392212815577088*Pi^5-828687819932172288*Pi^4+ 848441544573714432*Pi^3-547714863950987264*Pi^2+197576625989091328*Pi-\ 29917994859626496)/i^4*(1/i)^(1/2)/(3*Pi-8)^13+1/256*Pi*(379701131328*Pi^13-\ 13578413819553*Pi^12+218752219226448*Pi^11-2068657320726144*Pi^10+ 12776076005446656*Pi^9-54547132817977344*Pi^8+166420162034663424*Pi^7-\ 369700861343956992*Pi^6+601845811675922432*Pi^5-710220601607847936*Pi^4+ 583330273588936704*Pi^3-304093960266055680*Pi^2+82515890943295488*Pi-\ 6481964643123200)/(3*Pi-8)^14/i^5-1/32768*Pi^(3/2)*(1222412422254741*Pi^13-\ 33923305359382320*Pi^12+438407240333262912*Pi^11-3505918772208416256*Pi^10+ 19363014219956797440*Pi^9-77713205767036305408*Pi^8+231422659053979435008*Pi^7-\ 512105291003266596864*Pi^6+831100185203852705792*Pi^5-963268026986135552000*Pi^ 4+763770745055424282624*Pi^3-385723890689881669632*Pi^2+108015592541106733056* Pi-11637219523509092352)/i^5*(1/i)^(1/2)/(3*Pi-8)^15+O(1/i^6), (13170*Pi^4+ 30528*Pi^3-145152*Pi^2-233472*Pi-65536)*2^(1/2)/(3*Pi-8)^(9/2)+(51030*Pi^4+ 354402*Pi^3-1805760*Pi^2-145152*Pi+2322432)*2^(1/2)*Pi^(3/2)*(1/i)^(1/2)/(3*Pi-\ 8)^(11/2)+3/4*2^(1/2)*Pi*(3400677*Pi^5-16424244*Pi^4+9586176*Pi^3+47427072*Pi^2 -58318848*Pi-22937600)/(3*Pi-8)^(13/2)/i+3/2*2^(1/2)*Pi^(3/2)*(2934225*Pi^6-\ 12170331*Pi^5-31198926*Pi^4+251031456*Pi^3-464124672*Pi^2+218652672*Pi+52756480 )/i*(1/i)^(1/2)/(3*Pi-8)^(15/2)+3/64*2^(1/2)*Pi*(1186330131*Pi^7-12692100600*Pi ^6+55687961232*Pi^5-124240421376*Pi^4+138354739200*Pi^3-54872014848*Pi^2-\ 27800371200*Pi+18555600896)/(3*Pi-8)^(17/2)/i^2+3/16*2^(1/2)*Pi^(3/2)*( 206263260*Pi^8-3172233537*Pi^7+22143472848*Pi^6-86633783532*Pi^5+197223886080* Pi^4-244393216512*Pi^3+115671441408*Pi^2+36684562432*Pi-43467669504)/i^2*(1/i)^ (1/2)/(3*Pi-8)^(19/2)-1/512*2^(1/2)*Pi*(221558382027*Pi^9-3523366329588*Pi^8+ 23696523492912*Pi^7-86989177348416*Pi^6+183241875124224*Pi^5-200894671429632*Pi ^4+44534791798784*Pi^3+138610533728256*Pi^2-124062277828608*Pi+21330955075584)/ (3*Pi-8)^(21/2)/i^3-9/64*2^(1/2)*Pi^(3/2)*(7049050920*Pi^10-123892410573*Pi^9+ 946496444328*Pi^8-4038747981444*Pi^7+10139901180072*Pi^6-13672001006208*Pi^5+ 3743987051520*Pi^4+17556683587584*Pi^3-27616953237504*Pi^2+16907088429056*Pi-\ 3768900911104)/i^3*(1/i)^(1/2)/(3*Pi-8)^(23/2)-1/16384*2^(1/2)*Pi*( 20430982218861*Pi^11-335296640001456*Pi^10+2615740610964768*Pi^9-\ 13654533537144576*Pi^8+56390194704708864*Pi^7-191640363278819328*Pi^6+ 504217306866548736*Pi^5-948083530438017024*Pi^4+1184506163027247104*Pi^3-\ 889066357423865856*Pi^2+328890316146868224*Pi-30455372577767424)/(3*Pi-8)^(25/2 )/i^4+1/1024*2^(1/2)*Pi^(3/2)*(10659388486320*Pi^12-226059952467105*Pi^11+ 2051707730023620*Pi^10-10092319189010136*Pi^9+26500347481873392*Pi^8-\ 17275759082116944*Pi^7-126517223075544576*Pi^6+501604210195494912*Pi^5-\ 951735959264919552*Pi^4+1070757972046512128*Pi^3-710690882206040064*Pi^2+ 245629609020751872*Pi-29771293809180672)/i^4*(1/i)^(1/2)/(3*Pi-8)^(27/2)+1/ 131072*2^(1/2)*Pi*(9615618398620737*Pi^13-231897873554319636*Pi^12+ 2577233980700948448*Pi^11-17555819629741188480*Pi^10+82188363072956656896*Pi^9-\ 280402972553106662400*Pi^8+717993269025205911552*Pi^7-1391324026121373745152*Pi ^6+2020260364203646255104*Pi^5-2137438038145628110848*Pi^4+ 1564626794235748679680*Pi^3-721033767056761159680*Pi^2+170429881227279335424*Pi -9956297691837235200)/(3*Pi-8)^(29/2)/i^5-3/4096*2^(1/2)*Pi^(3/2)*( 62950121786160*Pi^14-1319513707621647*Pi^13+8837999116429068*Pi^12+ 14827686138188928*Pi^11-641310998346130032*Pi^10+5070901516480793232*Pi^9-\ 23177537154844970976*Pi^8+71260902360471654912*Pi^7-154964894249700519936*Pi^6+ 241660923510239133696*Pi^5-267415966497712898048*Pi^4+202273799806283415552*Pi^ 3-96493929729292763136*Pi^2+24343542727245299712*Pi-1936385318206832640)/i^5*(1 /i)^(1/2)/(3*Pi-8)^(31/2)+O(1/i^6), 3*(3465*Pi^5+62200*Pi^4-21120*Pi^3-494592* Pi^2-471040*Pi-98304)/(3*Pi-8)^5+(1626750*Pi^4-1106160*Pi^3-18051840*Pi^2+ 11704320*Pi+17203200)*Pi^(3/2)*(1/i)^(1/2)/(3*Pi-8)^6+15*Pi*(274995*Pi^6+916344 *Pi^5-12013632*Pi^4+21467136*Pi^3+5419008*Pi^2-27721728*Pi-6553600)/(3*Pi-8)^7/ i+15/4*Pi^(3/2)*(39874923*Pi^6-292323096*Pi^5+650382272*Pi^4-30887424*Pi^3-\ 1542610944*Pi^2+1211170816*Pi+145752064)/i*(1/i)^(1/2)/(3*Pi-8)^8+3/4*Pi*( 284143545*Pi^8-1997737830*Pi^7+1654218000*Pi^6+23292092032*Pi^5-88670665728*Pi^ 4+126643126272*Pi^3-65964474368*Pi^2-8124366848*Pi+6627000320)/(3*Pi-8)^9/i^2+3 /64*Pi^(3/2)*(42264235485*Pi^8-556699667400*Pi^7+3186362350080*Pi^6-\ 10172821954560*Pi^5+19316946309120*Pi^4-21253706776576*Pi^3+11232857292800*Pi^2 -688968957952*Pi-1227085578240)/i^2*(1/i)^(1/2)/(3*Pi-8)^10+1/8*Pi*(3261287205* Pi^10-104752787490*Pi^9+1298177753040*Pi^8-8254390499712*Pi^7+30281809185792*Pi ^6-66358159073280*Pi^5+83677588684800*Pi^4-50223267184640*Pi^3-986047315968*Pi^ 2+13453951500288*Pi-1904549560320)/(3*Pi-8)^11/i^3-5/512*Pi^(3/2)*( 2962656131949*Pi^10-53713233641736*Pi^9+426761792268480*Pi^8-1942460404379136* Pi^7+5519139155546112*Pi^6-9901116793847808*Pi^5+10525669198135296*Pi^4-\ 5029782691512320*Pi^3-981955856302080*Pi^2+1841520646815744*Pi-385271451353088) /i^3*(1/i)^(1/2)/(3*Pi-8)^12-1/128*Pi*(5366900466936*Pi^12-111356293021083*Pi^ 11+1046934575794728*Pi^10-5825153277893952*Pi^9+20984631631308288*Pi^8-\ 50183619675254784*Pi^7+77679401157033984*Pi^6-68524911458254848*Pi^5+ 12572235569037312*Pi^4+43253140283195392*Pi^3-48384368078487552*Pi^2+ 18806003931807744*Pi-1359614847221760)/(3*Pi-8)^13/i^4+1/16384*Pi^(3/2)*( 1465626345516261*Pi^12-37679884502621544*Pi^11+419190597331560576*Pi^10-\ 2673098265793987584*Pi^9+10727632849620934656*Pi^8-27414206448491495424*Pi^7+ 40803300727257563136*Pi^6-18698856772134764544*Pi^5-50250141231537455104*Pi^4+ 108437521231643672576*Pi^3-92446164193483161600*Pi^2+34828862613356544000*Pi-\ 3392374077590077440)/i^4*(1/i)^(1/2)/(3*Pi-8)^14+1/16*Pi*(10605311884512*Pi^14-\ 260400586731615*Pi^13+2860256763844920*Pi^12-18377323159706604*Pi^11+ 75332353862112480*Pi^10-198451751086259712*Pi^9+301967428366970880*Pi^8-\ 92015299292774400*Pi^7-697903583741214720*Pi^6+1763963105186938880*Pi^5-\ 2249598040820154368*Pi^4+1718707819265392640*Pi^3-755490470410321920*Pi^2+ 156084029274193920*Pi-6944962117632000)/(3*Pi-8)^15/i^5+1/131072*Pi^(3/2)*( 363073114458974907*Pi^14-9292478157285751512*Pi^13+109975663793974691520*Pi^12-\ 802009043877679879680*Pi^11+4062572219921041981440*Pi^10-\ 15307675778431331205120*Pi^9+44782669423650173091840*Pi^8-\ 103877301379203826974720*Pi^7+190553427164491646238720*Pi^6-\ 269578352459267985899520*Pi^5+281574482435066537443328*Pi^4-\ 204098935423993588482048*Pi^3+93247688953331491799040*Pi^2-\ 22024236420456995880960*Pi+1326510553220587192320)/i^5*(1/i)^(1/2)/(3*Pi-8)^16+ O(1/i^6)] and in floating point [0., 1., -.4856928233+1.337342216*(1/i)^(1/2)+.7922638422/i+.2113312509/i*(1/i) ^(1/2)+.1363587842/i^2+.3926972959/i^2*(1/i)^(1/2)+.5294326428/i^3+.3202072697/ i^3*(1/i)^(1/2)-.7136088458e-1/i^4+1.196118854/i^4*(1/i)^(1/2)+85.91739013/i^5-\ 715.8143186/i^5*(1/i)^(1/2)+O(1/i^6), 3.108163850-2.521611529*(1/i)^(1/2)+.7572\ 348995/i+2.123245844/i*(1/i)^(1/2)+.3550629074/i^2-1.579105648/i^2*(1/i)^(1/2)-\ .9203763402/i^3+1.402441457/i^3*(1/i)^(1/2)-.4414278227/i^4+2.906264130/i^4*(1/ i)^(1/2)+574.3213271/i^5+539.4300446/i^5*(1/i)^(1/2)+O(1/i^6), -4.642979749+15.\ 83991694*(1/i)^(1/2)-1.522081340/i-5.233446531/i*(1/i)^(1/2)+5.020947150/i^2+8.\ 094931938/i^2*(1/i)^(1/2)-4.084060574/i^3-16.48391111/i^3*(1/i)^(1/2)+38.618647\ 45/i^4+52.95487194/i^4*(1/i)^(1/2)-6241.247616/i^5-98586.22734/i^5*(1/i)^(1/2)+ O(1/i^6), 18.66866547-49.77979646*(1/i)^(1/2)+47.96251716/i+33.76060470/i*(1/i) ^(1/2)-39.36631622/i^2-30.85375841/i^2*(1/i)^(1/2)+50.12016322/i^3+44.48381961/ i^3*(1/i)^(1/2)-1306.250313/i^4-4676.973284/i^4*(1/i)^(1/2)-179946.6738/i^5-\ 603527.2993/i^5*(1/i)^(1/2)+O(1/i^6), -48.55583951+231.4004015*(1/i)^(1/2)-266.\ 8243333/i-10.20470774/i*(1/i)^(1/2)+327.0117572/i^2-17.77863605/i^2*(1/i)^(1/2) -491.8642865/i^3-49.33990276/i^3*(1/i)^(1/2)+2557.863621/i^4-65593.44943/i^4*(1 /i)^(1/2)-1480961.164/i^5+45036393.02/i^5*(1/i)^(1/2)+O(1/i^6), 181.0912166-925\ .8379181*(1/i)^(1/2)+1763.027322/i-601.7700651/i*(1/i)^(1/2)-1767.353824/i^2+ 1392.804168/i^2*(1/i)^(1/2)+2393.393175/i^3-4145.163008/i^3*(1/i)^(1/2)+19781.2\ 0480/i^4+533805.6555/i^4*(1/i)^(1/2)-3904700.216/i^5-61399185.90/i^5*(1/i)^(1/2 )+O(1/i^6), -622.5721500+4223.883677*(1/i)^(1/2)-10070.07390/i+8558.585175/i*(1 /i)^(1/2)+7481.998973/i^2-16163.00954/i^2*(1/i)^(1/2)-584.1564693/i^3+10672.302\ 16/i^3*(1/i)^(1/2)+99730.10498/i^4-1186384.520/i^4*(1/i)^(1/2)-234931153.9/i^5-\ 2870286853./i^5*(1/i)^(1/2)+O(1/i^6), 2454.604104-19155.85447*(1/i)^(1/2)+59293\ .94750/i-76310.70004/i*(1/i)^(1/2)-8035.132125/i^2+136480.1445/i^2*(1/i)^(1/2)-\ 1095.371341/i^3-1428560.363/i^3*(1/i)^(1/2)-2345460.886/i^4-53713062.39/i^4*(1/ i)^(1/2)+760397380.6/i^5-.1529469721e11/i^5*(1/i)^(1/2)+O(1/i^6)] --------------------------------------------------------------- This ends this article that took, 133.240, seconds to generate. ----------------------- This took, 133.344, seconds.