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 37281 10080 + ----------------------------- - ----- - ----- - ----- 5/2 1/2 2 2 Pi 3 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^2-37281/2/Pi -10080/Pi^3+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 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) 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] and in floating point [0., 1., -.4856928234, 3.108163850, -4.642979574, 18.66866547] 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 3/2 1/2 1/2 2 (102 Pi - 160 Pi - 512) 2 225 Pi (1/i) 2 (Pi - 2 Pi - 32/9) ----------------------------- + ------------------------------------------- 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 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*Pi^(3/2)*(1/i)^(1/2)*2^(1/2)*(Pi^2-2*Pi-32/9)/(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)] 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.\ 83991693*(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)] --------------------------------------------------------------- This ends this article that took, 7.169, seconds to generate. ----------------------- This took, 7.273, seconds.