The first , 5, positive terms and the first , 5, negative terms in the exact formula for the "Big Guys" aproximations to 1 / | (n - r) | (P[k](exp(2 I Pi t)) P[k](exp(-2 I Pi t))) | / 0 r (P[k](-exp(2 I Pi t)) P[k](-exp(-2 I Pi t))) dt n n k / 2 r! (n - r)! (2 ) n |27 (n - 1) (n - 2) (n - 3) n gu := -------------------- + 2 r! (n - r)! |---------------------------- (n + 1)! \ 8 2 3 135 n (2 n - 3 n + 5) r 2 2 945 n r - ------------------------ + (1215/4 n - 405/4 n + 675/4) r - -------- 4 2 4\ / 945 r | (n - 4) k n | + ------| (6 2 ) /(n + 5)! + 2 r! (n - r)! | 4 / \ 595 n (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (n - 7) ------------------------------------------------------------- 128 6 5 4 3 2 1785 n (6 n - 63 n + 637 n - 1960 n + 6083 n - 4949 n + 4566) r / - -------------------------------------------------------------------- + | 32 \ 77381535 2 6559875 3 4075155 187425 6 937125 5 17805375 4 -------- n - ------- n + ------- + ------ n - ------ n + -------- n 64 16 16 32 32 64 4 3 2 3 8833965 \ 2 687225 n (2 n - 5 n + 46 n - 28 n + 81) r - ------- n| r - --------------------------------------------- 32 / 32 /10308375 4 6185025 3 112017675 2 4810575 55665225\ 4 + |-------- n - ------- n + --------- n - ------- n + --------| r \ 64 32 64 16 64 / 2 5 5360355 n (2 n - n + 9) r /12507495 2 1786785 16081065\ 6 - --------------------------- + |-------- n - ------- n + --------| r 32 \ 32 32 32 / 7 8\ / 3828825 n r 3828825 r | (n - 8) k n | - ------------ + ----------| (70 2 ) /(n + 9)! + 2 r! (n - r)! | 16 64 / \ 5775 n (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (n - 7) (n - 8) 10 9 8 (n - 9) (n - 10) (n - 11)/1024 - 75075 n (6 n - 165 n + 4235 n 7 6 5 4 3 - 43560 n + 429132 n - 2004849 n + 9223335 n - 18770730 n 2 /17342325 10 + 39343612 n - 25465176 n + 19056960) r/512 + |-------- n \ 512 2490479266275 2 1245562943625 3 370380035025 6 + 22354707375/8 + ------------- n - ------------- n + ------------ n 128 128 512 1061610424875 5 2303519092875 4 238974761025 20030385375 7 - ------------- n + ------------- n - ------------ n - ----------- n 512 256 64 256 780404625 8 260134875 9\ 2 8 7 6 + --------- n - --------- n | r - 5780775 n (25 n - 225 n + 5190 n 64 512 / 5 4 3 2 3 - 20790 n + 184701 n - 307395 n + 1288240 n - 618090 n + 1212328) r / /25109171962875 2 3053203027875 3 876024424275 81942485625 6 256 + |-------------- n - ------------- n + ------------ + ----------- n \ 512 256 64 128 100151926875 5 6941699139375 4 1786519609875 1820944125 7 - ------------ n + ------------- n - ------------- n - ---------- n 64 512 256 64 1300674375 8\ 4 + ---------- n | r - 294819525 n 256 / 6 5 4 3 2 5 / (24 n - 84 n + 1860 n - 2640 n + 22526 n - 8811 n + 35945) r /256 + | \ 3532429275375 926028128025 2 437806994625 3 6191210025 6 ------------- + ------------ n - ------------ n + ---------- n 256 32 128 64 13266878625 5 1163063026125 4 865884944925 \ 6 - ----------- n + ------------- n - ------------ n| r 64 256 256 / 4 3 2 7 800224425 n (36 n - 45 n + 980 n - 330 n + 2782) r / - ------------------------------------------------------ + | 128 \ 180050495625 4 60016831875 3 2604730503375 2 132037030125 ------------ n - ----------- n + ------------- n - ------------ n 512 256 512 256 2 9 1113112175175\ 8 9335951625 n (10 n - 3 n + 65) r + -------------| r - ---------------------------------- 256 / 256 11 /61617280725 2 5601570975 121367371125\ 10 11712375675 n r + |----------- n - ---------- n + ------------| r - ----------------- \ 256 256 256 / 128 12\ / 3904125225 r | (n - 12) k n | + --------------| (924 2 ) /(n + 13)! + 2 r! (n - r)! |212355 256 / \ (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (n - 7) (n - 8) (n - 9) (n - 10) (n - 11) (n - 12) (n - 13) (n - 14) (n - 15) n/32768 - 3610035 n ( 14 13 12 11 10 9 2 n - 105 n + 5075 n - 109200 n + 2215577 n - 24954930 n 8 7 6 5 + 269790235 n - 1691718600 n + 10363418065 n - 35376706365 n 4 3 2 + 120835413790 n - 199949349600 n + 345786211656 n - 195883887600 n /371508005113245 10 14783093325 13 + 130026254400) r/4096 + |--------------- n - ----------- n \ 8192 4096 1026716033800758015 2 1335406097025 12 7334364520357875 + ------------------- n + ------------- n + ---------------- 1024 8192 64 2439210398625 11 310050768641488875 3 487354725 14 - ------------- n - ------------------ n + --------- n 1024 512 4096 756310216422946185 6 839901935101954725 5 2719847483007229575 4 + ------------------ n - ------------------ n + ------------------- n 8192 4096 4096 44196730635754125 16340270460388875 7 30018015002514825 8 - ----------------- n - ----------------- n + ----------------- n 256 1024 8192 737129382464475 9\ 2 12 11 10 - --------------- n | r - 342953325 n (42 n - 819 n + 35490 n 2048 / 9 8 7 6 5 - 360360 n + 6573567 n - 36540504 n + 356791292 n - 1047506460 n 4 3 2 + 5826890069 n - 7862872017 n + 24617748428 n - 10255518000 n 3 /260561875248675 10 13910078982259134675 2 + 16670279016) r /4096 + |--------------- n + -------------------- n \ 8192 4096 468131288625 12 3089666504925 11 4528840372140909825 3 + ------------ n - ------------- n - ------------------- n 8192 4096 4096 714640952151866025 298392095231255025 6 282925911300157125 5 + ------------------ + ------------------ n - ------------------ n 1024 2048 1024 24523650801798323175 4 219822749856084375 31468253352661125 7 + -------------------- n - ------------------ n - ----------------- n 16384 512 2048 65652323563151325 8 231724987869375 9\ 4 10 + ----------------- n - --------------- n | r - 18725251545 n (126 n 16384 1024 / 9 8 7 6 5 4 - 1155 n + 47795 n - 237600 n + 4119698 n - 10619070 n + 98656580 n 3 2 5 / - 113384700 n + 603937851 n - 203823675 n + 620896950) r /4096 + | \ 15860288058615 10 1937741928712214625 24708604555434654855 2 -------------- n + ------------------- + -------------------- n 4096 2048 8192 585538147680861375 3 243820208325088395 6 202932385709978925 5 - ------------------ n + ------------------ n - ------------------ n 1024 4096 2048 3718552673669959125 4 1272216528400442625 3568564813188375 7 + ------------------- n - ------------------- n - ---------------- n 4096 4096 1024 8326651230772875 8 101958994662525 9\ 6 8 + ---------------- n - --------------- n | r - 3383920457775 n (22 n 8192 4096 / 7 6 5 4 3 2 - 99 n + 4004 n - 9240 n + 156170 n - 157836 n + 1433116 n 7 /7358291044465786425 - 414000 n + 2174487) r /4096 + |------------------- \ 16384 7899873978050844075 2 218658788220047175 3 62795411934930675 6 + ------------------- n - ------------------ n + ----------------- n 8192 2048 8192 11725284386190375 5 3147800639992833825 4 87558941844928125 - ----------------- n + ------------------- n - ----------------- n 1024 16384 1024 781685625746025 7 1005024375959175 8\ 8 - --------------- n + ---------------- n | r - 5639867429625 n 4096 16384 / 6 5 4 3 2 9 (110 n - 231 n + 9233 n - 8400 n + 140273 n - 35910 n + 323595) r / /365006580177900375 1025579436793186275 2 7698419041438125 3 4096 + |------------------ + ------------------- n - ---------------- n \ 4096 8192 1024 1129101459410925 6 1539683808287625 5 122661476726914125 4 + ---------------- n - ---------------- n + ------------------ n 4096 4096 8192 20252763939783375 \ 10 - ----------------- n| r 2048 / 4 3 2 11 6460211783025 n (234 n - 195 n + 7750 n - 1800 n + 29937) r / - ---------------------------------------------------------------- + | 4096 \ 2939396361276375 4 678322237217625 3 53845865211513375 2 ---------------- n - --------------- n + ----------------- n 8192 4096 8192 484515883726875 64466453382806475\ 12 - --------------- n + -----------------| r 1024 8192 / 2 13 72056208349125 n (14 n - 3 n + 119) r - ---------------------------------------- 4096 /463218482244375 2 30881232149625 1224955541935125\ 14 + |--------------- n - -------------- n + ----------------| r \ 4096 4096 4096 / 15 16\ 63821213109225 n r 63821213109225 r | (n - 16) k - -------------------- + ------------------| (12870 2 ) /(n + 17)! 2048 16384 / / 2 \ n | 5 n (n - 1) 15 n r 15 r | + 2 r! (n - r)! |- ----------------- + --------------- - ---------------| \ 2 (n + 3) (n + 2) (n + 3) (n + 2) (n + 3) (n + 2)/ / (n - 2) k n | (-2 2 ) /(n + 1)! + 2 r! (n - r)! | \ 65 n (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) - -------------------------------------------- 16 (n + 7) (n + 6) 4 3 2 1365 n (n - 5 n + 25 n - 25 n + 28) r + ---------------------------------------- 8 (n + 7) (n + 6) 4 3 2 2 1365 (10 n - 20 n + 95 n - 25 n + 28) r - ------------------------------------------- 8 (n + 7) (n + 6) 2 3 2 4 6825 n (4 n - 3 n + 14) r 6825 (15 n - 3 n + 14) r + --------------------------- - -------------------------- 4 (n + 7) (n + 6) 8 (n + 7) (n + 6) 5 6 \ 45045 n r 15015 r | (n - 6) k n + ----------------- - -----------------| (-20 2 ) /(n + 5)! + 2 r! 4 (n + 7) (n + 6) 4 (n + 7) (n + 6)/ / | (n - r)! |- 1323 (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (n - 7) \ 8 7 6 (n - 8) (n - 9) n/(256 (n + 11) (n + 10)) + 14553 n (5 n - 90 n + 1530 n 5 4 3 2 - 9450 n + 56847 n - 136080 n + 332290 n - 236700 n + 193248) r/(128 8 7 6 5 4 (n + 10) (n + 11)) - 43659 (45 n - 420 n + 6685 n - 23625 n + 133700 n 3 2 2 - 164115 n + 381510 n - 78900 n + 64416) r /(128 (n + 10) (n + 11)) + 6 5 4 3 2 3 945945 n (12 n - 63 n + 966 n - 1890 n + 10332 n - 5481 n + 12496) r / (64 (n + 10) (n + 11)) - 945945 6 5 4 3 2 4 (147 n - 441 n + 6615 n - 6615 n + 35511 n - 5481 n + 12496) r /(128 4 3 2 5 3972969 n (63 n - 105 n + 1555 n - 675 n + 3597) r (n + 10) (n + 11)) + ------------------------------------------------------ 64 (n + 10) (n + 11) 4 3 2 6 3972969 (140 n - 120 n + 1765 n - 225 n + 1199) r - ----------------------------------------------------- 64 (n + 10) (n + 11) 2 7 2 8 48243195 n (8 n - 3 n + 44) r 48243195 (27 n - 3 n + 44) r + ------------------------------- - ------------------------------ 32 (n + 10) (n + 11) 128 (n + 10) (n + 11) 9 10 \ 305540235 n r 61108047 r | (n - 10) k + -------------------- - --------------------| (-252 2 ) /(n + 9)! 64 (n + 11) (n + 10) 64 (n + 11) (n + 10)/ / n | + 2 r! (n - r)! |- 12441 n (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) \ (n - 6) (n - 7) (n - 8) (n - 9) (n - 10) (n - 11) (n - 12) (n - 13)/(2048 12 11 10 9 (n + 15) (n + 14)) + 12441 n (105 n - 4095 n + 147875 n - 2277275 n 8 7 6 5 + 33270237 n - 253870617 n + 1876434417 n - 7307437137 n 4 3 2 + 28271361118 n - 51184761628 n + 96204101448 n - 57836888928 n 12 11 + 40495092480) r/(1024 (n + 15) (n + 14)) - 37323 (910 n - 20020 n 10 9 8 7 6 + 675675 n - 6831825 n + 93633540 n - 478227750 n + 3328249925 n 5 4 3 2 - 8247314075 n + 30178486338 n - 29673621978 n + 53306197308 n 2 - 9639481488 n + 6749182080) r /(512 (n + 15) (n + 14)) + 96231135 n ( 10 9 8 7 6 5 4 4 n - 55 n + 1782 n - 12012 n + 157872 n - 528528 n + 3534223 n 3 2 3 - 5226793 n + 18478339 n - 8200192 n + 14455920) r /(256 (n + 15) 10 9 8 7 6 (n + 14)) - 96231135 (198 n - 1782 n + 56232 n - 252252 n + 3223143 n 5 4 3 2 - 6819813 n + 44424897 n - 35458137 n + 122968538 n - 16400384 n 4 8 7 + 28911840) r /(1024 (n + 15) (n + 14)) + 12067384329 n (6 n - 36 n 6 5 4 3 2 + 1116 n - 3276 n + 41069 n - 51779 n + 331799 n - 119639 n + 411540) 5 8 7 6 r /(512 (n + 15) (n + 14)) - 4022461443 (90 n - 360 n + 11025 n 5 4 3 2 6 - 20475 n + 253449 n - 172809 n + 1097099 n - 119639 n + 411540) r /( 512 (n + 15) (n + 14)) + 8619560235 n 6 5 4 3 2 7 (72 n - 189 n + 5740 n - 6370 n + 78218 n - 24206 n + 153060) r /(256 (n + 15) (n + 14)) - 8619560235 6 5 4 3 2 8 (693 n - 1155 n + 34881 n - 21021 n + 256928 n - 24206 n + 153060) r / (1024 (n + 15) (n + 14)) 4 3 2 9 462583065945 n (11 n - 11 n + 331 n - 91 n + 1110) r + -------------------------------------------------------- 512 (n + 15) (n + 14) 4 3 2 10 92516613189 (66 n - 36 n + 1081 n - 91 n + 1110) r - ------------------------------------------------------- 512 (n + 15) (n + 14) 2 11 2 12 630795089925 n (4 n - n + 30) r 210265029975 (13 n - n + 30) r + ---------------------------------- - --------------------------------- 256 (n + 15) (n + 14) 512 (n + 15) (n + 14) 13 14 \ 436704293025 n r 62386327575 r | (n - 14) k + --------------------- - ---------------------| (-3432 2 ) / 256 (n + 15) (n + 14) 256 (n + 15) (n + 14)/ / n | (n + 13)! + 2 r! (n - r)! |- 449735 n (n - 1) (n - 2) (n - 3) (n - 4) \ (n - 5) (n - 6) (n - 7) (n - 8) (n - 9) (n - 10) (n - 11) (n - 12) (n - 13) (n - 14) (n - 15) (n - 16) (n - 17)/(65536 (n + 19) (n + 18)) + 25634895 n 16 15 14 13 12 11 (3 n - 204 n + 12716 n - 364140 n + 9791558 n - 152429004 n 10 9 8 7 + 2264838056 n - 20737062060 n + 183797955055 n - 999143042040 n 6 5 4 + 5342097893860 n - 16475066407800 n + 51143533311184 n 3 2 - 78728553219552 n + 127381990200768 n - 68681531577600 n 16 + 43619859763200) r/(32768 (n + 19) (n + 18)) - 25634895 (255 n 15 14 13 12 11 - 10200 n + 594150 n - 11834550 n + 298744264 n - 3389888502 n 10 9 8 7 + 47462936378 n - 317791818630 n + 2661258794337 n - 10276480840626 n 6 5 4 + 52028971928024 n - 105975701706420 n + 312551494925344 n 3 2 - 267973692419472 n + 415749480378048 n - 68681531577600 n 2 14 + 43619859763200) r /(32768 (n + 19) (n + 18)) + 2178966075 n (28 n 13 12 11 10 9 - 735 n + 41062 n - 584766 n + 14139580 n - 117162045 n 8 7 6 5 + 1572416131 n - 7586878728 n + 60986833622 n - 161690667138 n 4 3 2 + 787776405689 n - 987114134916 n + 2816911773328 n - 1113206701176 n 3 14 + 1696279353984) r /(8192 (n + 19) (n + 18)) - 2178966075 (2310 n 13 12 11 10 9 - 42042 n + 2284282 n - 23585562 n + 553159607 n - 3330618291 n 8 7 6 5 + 43346298996 n - 147519828456 n + 1151336447261 n - 2000942260317 n 4 3 2 + 9495344585272 n - 6580626366132 n + 18426002361456 n 4 - 2226413402352 n + 3392558707968) r /(32768 (n + 19) (n + 18)) + 12 11 10 9 8 2315369351295 n (14 n - 182 n + 9698 n - 72930 n + 1672957 n 7 6 5 4 3 - 7234656 n + 92060644 n - 214352320 n + 1638099229 n - 1746159012 n 2 5 + 8145600658 n - 2601015300 n + 7223138800) r /(16384 (n + 19) (n + 18)) 12 11 10 9 8 - 771789783765 (364 n - 3432 n + 180323 n - 984555 n + 22219197 n 7 6 5 4 - 67693626 n + 847365974 n - 1292180370 n + 9731931664 n 3 2 6 - 5740757217 n + 26513943378 n - 2601015300 n + 7223138800) r /(16384 10 9 8 (n + 19) (n + 18)) + 21499858262025 n (40 n - 275 n + 14300 n 7 6 5 4 3 - 56100 n + 1250844 n - 2608242 n + 32265158 n - 30212740 n 2 7 + 225374304 n - 61455357 n + 282589888) r /(8192 (n + 19) (n + 18)) - 10 9 8 7 6 21499858262025 (715 n - 3575 n + 184470 n - 510510 n + 11281725 n 5 4 3 2 - 15431325 n + 189318830 n - 98366590 n + 729643329 n - 61455357 n 8 8 + 282589888) r /(32768 (n + 19) (n + 18)) + 35833097103375 n (715 n 7 6 5 4 3 2 - 2574 n + 132054 n - 250614 n + 5502567 n - 4633146 n + 56530604 n 9 - 13630770 n + 100853805) r /(16384 (n + 19) (n + 18)) - 21499858262025 ( 8 7 6 5 4 3 3003 n - 7644 n + 390481 n - 487305 n + 10650682 n - 4991931 n 2 10 + 60709984 n - 4543590 n + 33617935) r /(16384 (n + 19) (n + 18)) + 396770111562825 n 6 5 4 3 2 11 (156 n - 273 n + 13902 n - 10710 n + 233368 n - 50949 n + 618716) r / (8192 (n + 19) (n + 18)) - 132256703854275 6 5 4 3 2 (1365 n - 1575 n + 80025 n - 34425 n + 748743 n - 50949 n + 618716) 12 r /(16384 (n + 19) (n + 18)) 4 3 2 13 4730720560941375 n (21 n - 15 n + 761 n - 153 n + 3325) r + -------------------------------------------------------------- 8192 (n + 19) (n + 18) 4 3 2 14 675817222991625 (120 n - 48 n + 2433 n - 153 n + 3325) r - ------------------------------------------------------------- 8192 (n + 19) (n + 18) 2 15 1486797890581575 n (16 n - 3 n + 152) r + ------------------------------------------ 4096 (n + 19) (n + 18) 2 16 17 1486797890581575 (51 n - 3 n + 152) r 9183163441827375 n r - ---------------------------------------- + ----------------------- 32768 (n + 19) (n + 18) 16384 (n + 19) (n + 18) 18 \ 1020351493536375 r | (n - 18) k - -----------------------| (-48620 2 ) /(n + 17)! 16384 (n + 19) (n + 18)/ n n k / 2 r! (n - r)! (2 ) n |27 (n - 1) (n - 2) (n - 3) n -------------------- + 2 r! (n - r)! |---------------------------- (n + 1)! \ 8 2 3 135 n (2 n - 3 n + 5) r 2 2 945 n r - ------------------------ + (1215/4 n - 405/4 n + 675/4) r - -------- 4 2 4\ / 945 r | (n - 4) k n | + ------| (6 2 ) /(n + 5)! + 2 r! (n - r)! | 4 / \ 595 n (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (n - 7) ------------------------------------------------------------- 128 6 5 4 3 2 1785 n (6 n - 63 n + 637 n - 1960 n + 6083 n - 4949 n + 4566) r / - -------------------------------------------------------------------- + | 32 \ 77381535 2 6559875 3 4075155 187425 6 937125 5 17805375 4 -------- n - ------- n + ------- + ------ n - ------ n + -------- n 64 16 16 32 32 64 4 3 2 3 8833965 \ 2 687225 n (2 n - 5 n + 46 n - 28 n + 81) r - ------- n| r - --------------------------------------------- 32 / 32 /10308375 4 6185025 3 112017675 2 4810575 55665225\ 4 + |-------- n - ------- n + --------- n - ------- n + --------| r \ 64 32 64 16 64 / 2 5 5360355 n (2 n - n + 9) r /12507495 2 1786785 16081065\ 6 - --------------------------- + |-------- n - ------- n + --------| r 32 \ 32 32 32 / 7 8\ / 3828825 n r 3828825 r | (n - 8) k n | - ------------ + ----------| (70 2 ) /(n + 9)! + 2 r! (n - r)! | 16 64 / \ 5775 n (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (n - 7) (n - 8) 10 9 8 (n - 9) (n - 10) (n - 11)/1024 - 75075 n (6 n - 165 n + 4235 n 7 6 5 4 3 - 43560 n + 429132 n - 2004849 n + 9223335 n - 18770730 n 2 /17342325 10 + 39343612 n - 25465176 n + 19056960) r/512 + |-------- n \ 512 2490479266275 2 1245562943625 3 370380035025 6 + 22354707375/8 + ------------- n - ------------- n + ------------ n 128 128 512 1061610424875 5 2303519092875 4 238974761025 20030385375 7 - ------------- n + ------------- n - ------------ n - ----------- n 512 256 64 256 780404625 8 260134875 9\ 2 8 7 6 + --------- n - --------- n | r - 5780775 n (25 n - 225 n + 5190 n 64 512 / 5 4 3 2 3 - 20790 n + 184701 n - 307395 n + 1288240 n - 618090 n + 1212328) r / /25109171962875 2 3053203027875 3 876024424275 81942485625 6 256 + |-------------- n - ------------- n + ------------ + ----------- n \ 512 256 64 128 100151926875 5 6941699139375 4 1786519609875 1820944125 7 - ------------ n + ------------- n - ------------- n - ---------- n 64 512 256 64 1300674375 8\ 4 + ---------- n | r - 294819525 n 256 / 6 5 4 3 2 5 / (24 n - 84 n + 1860 n - 2640 n + 22526 n - 8811 n + 35945) r /256 + | \ 3532429275375 926028128025 2 437806994625 3 6191210025 6 ------------- + ------------ n - ------------ n + ---------- n 256 32 128 64 13266878625 5 1163063026125 4 865884944925 \ 6 - ----------- n + ------------- n - ------------ n| r 64 256 256 / 4 3 2 7 800224425 n (36 n - 45 n + 980 n - 330 n + 2782) r / - ------------------------------------------------------ + | 128 \ 180050495625 4 60016831875 3 2604730503375 2 132037030125 ------------ n - ----------- n + ------------- n - ------------ n 512 256 512 256 2 9 1113112175175\ 8 9335951625 n (10 n - 3 n + 65) r + -------------| r - ---------------------------------- 256 / 256 11 /61617280725 2 5601570975 121367371125\ 10 11712375675 n r + |----------- n - ---------- n + ------------| r - ----------------- \ 256 256 256 / 128 12\ / 3904125225 r | (n - 12) k n | + --------------| (924 2 ) /(n + 13)! + 2 r! (n - r)! |212355 256 / \ (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (n - 7) (n - 8) (n - 9) (n - 10) (n - 11) (n - 12) (n - 13) (n - 14) (n - 15) n/32768 - 3610035 n ( 14 13 12 11 10 9 2 n - 105 n + 5075 n - 109200 n + 2215577 n - 24954930 n 8 7 6 5 + 269790235 n - 1691718600 n + 10363418065 n - 35376706365 n 4 3 2 + 120835413790 n - 199949349600 n + 345786211656 n - 195883887600 n /371508005113245 10 14783093325 13 + 130026254400) r/4096 + |--------------- n - ----------- n \ 8192 4096 1026716033800758015 2 1335406097025 12 7334364520357875 + ------------------- n + ------------- n + ---------------- 1024 8192 64 2439210398625 11 310050768641488875 3 487354725 14 - ------------- n - ------------------ n + --------- n 1024 512 4096 756310216422946185 6 839901935101954725 5 2719847483007229575 4 + ------------------ n - ------------------ n + ------------------- n 8192 4096 4096 44196730635754125 16340270460388875 7 30018015002514825 8 - ----------------- n - ----------------- n + ----------------- n 256 1024 8192 737129382464475 9\ 2 12 11 10 - --------------- n | r - 342953325 n (42 n - 819 n + 35490 n 2048 / 9 8 7 6 5 - 360360 n + 6573567 n - 36540504 n + 356791292 n - 1047506460 n 4 3 2 + 5826890069 n - 7862872017 n + 24617748428 n - 10255518000 n 3 /260561875248675 10 13910078982259134675 2 + 16670279016) r /4096 + |--------------- n + -------------------- n \ 8192 4096 468131288625 12 3089666504925 11 4528840372140909825 3 + ------------ n - ------------- n - ------------------- n 8192 4096 4096 714640952151866025 298392095231255025 6 282925911300157125 5 + ------------------ + ------------------ n - ------------------ n 1024 2048 1024 24523650801798323175 4 219822749856084375 31468253352661125 7 + -------------------- n - ------------------ n - ----------------- n 16384 512 2048 65652323563151325 8 231724987869375 9\ 4 10 + ----------------- n - --------------- n | r - 18725251545 n (126 n 16384 1024 / 9 8 7 6 5 4 - 1155 n + 47795 n - 237600 n + 4119698 n - 10619070 n + 98656580 n 3 2 5 / - 113384700 n + 603937851 n - 203823675 n + 620896950) r /4096 + | \ 15860288058615 10 1937741928712214625 24708604555434654855 2 -------------- n + ------------------- + -------------------- n 4096 2048 8192 585538147680861375 3 243820208325088395 6 202932385709978925 5 - ------------------ n + ------------------ n - ------------------ n 1024 4096 2048 3718552673669959125 4 1272216528400442625 3568564813188375 7 + ------------------- n - ------------------- n - ---------------- n 4096 4096 1024 8326651230772875 8 101958994662525 9\ 6 8 + ---------------- n - --------------- n | r - 3383920457775 n (22 n 8192 4096 / 7 6 5 4 3 2 - 99 n + 4004 n - 9240 n + 156170 n - 157836 n + 1433116 n 7 /7358291044465786425 - 414000 n + 2174487) r /4096 + |------------------- \ 16384 7899873978050844075 2 218658788220047175 3 62795411934930675 6 + ------------------- n - ------------------ n + ----------------- n 8192 2048 8192 11725284386190375 5 3147800639992833825 4 87558941844928125 - ----------------- n + ------------------- n - ----------------- n 1024 16384 1024 781685625746025 7 1005024375959175 8\ 8 - --------------- n + ---------------- n | r - 5639867429625 n 4096 16384 / 6 5 4 3 2 9 (110 n - 231 n + 9233 n - 8400 n + 140273 n - 35910 n + 323595) r / /365006580177900375 1025579436793186275 2 7698419041438125 3 4096 + |------------------ + ------------------- n - ---------------- n \ 4096 8192 1024 1129101459410925 6 1539683808287625 5 122661476726914125 4 + ---------------- n - ---------------- n + ------------------ n 4096 4096 8192 20252763939783375 \ 10 - ----------------- n| r 2048 / 4 3 2 11 6460211783025 n (234 n - 195 n + 7750 n - 1800 n + 29937) r / - ---------------------------------------------------------------- + | 4096 \ 2939396361276375 4 678322237217625 3 53845865211513375 2 ---------------- n - --------------- n + ----------------- n 8192 4096 8192 484515883726875 64466453382806475\ 12 - --------------- n + -----------------| r 1024 8192 / 2 13 72056208349125 n (14 n - 3 n + 119) r - ---------------------------------------- 4096 /463218482244375 2 30881232149625 1224955541935125\ 14 + |--------------- n - -------------- n + ----------------| r \ 4096 4096 4096 / 15 16\ 63821213109225 n r 63821213109225 r | (n - 16) k - -------------------- + ------------------| (12870 2 ) /(n + 17)! 2048 16384 / / 2 \ n | 5 n (n - 1) 15 n r 15 r | + 2 r! (n - r)! |- ----------------- + --------------- - ---------------| \ 2 (n + 3) (n + 2) (n + 3) (n + 2) (n + 3) (n + 2)/ / (n - 2) k n | (-2 2 ) /(n + 1)! + 2 r! (n - r)! | \ 65 n (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) - -------------------------------------------- 16 (n + 7) (n + 6) 4 3 2 1365 n (n - 5 n + 25 n - 25 n + 28) r + ---------------------------------------- 8 (n + 7) (n + 6) 4 3 2 2 1365 (10 n - 20 n + 95 n - 25 n + 28) r - ------------------------------------------- 8 (n + 7) (n + 6) 2 3 2 4 6825 n (4 n - 3 n + 14) r 6825 (15 n - 3 n + 14) r + --------------------------- - -------------------------- 4 (n + 7) (n + 6) 8 (n + 7) (n + 6) 5 6 \ 45045 n r 15015 r | (n - 6) k n + ----------------- - -----------------| (-20 2 ) /(n + 5)! + 2 r! 4 (n + 7) (n + 6) 4 (n + 7) (n + 6)/ / | (n - r)! |- 1323 (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (n - 7) \ 8 7 6 (n - 8) (n - 9) n/(256 (n + 11) (n + 10)) + 14553 n (5 n - 90 n + 1530 n 5 4 3 2 - 9450 n + 56847 n - 136080 n + 332290 n - 236700 n + 193248) r/(128 8 7 6 5 4 (n + 10) (n + 11)) - 43659 (45 n - 420 n + 6685 n - 23625 n + 133700 n 3 2 2 - 164115 n + 381510 n - 78900 n + 64416) r /(128 (n + 10) (n + 11)) + 6 5 4 3 2 3 945945 n (12 n - 63 n + 966 n - 1890 n + 10332 n - 5481 n + 12496) r / (64 (n + 10) (n + 11)) - 945945 6 5 4 3 2 4 (147 n - 441 n + 6615 n - 6615 n + 35511 n - 5481 n + 12496) r /(128 4 3 2 5 3972969 n (63 n - 105 n + 1555 n - 675 n + 3597) r (n + 10) (n + 11)) + ------------------------------------------------------ 64 (n + 10) (n + 11) 4 3 2 6 3972969 (140 n - 120 n + 1765 n - 225 n + 1199) r - ----------------------------------------------------- 64 (n + 10) (n + 11) 2 7 2 8 48243195 n (8 n - 3 n + 44) r 48243195 (27 n - 3 n + 44) r + ------------------------------- - ------------------------------ 32 (n + 10) (n + 11) 128 (n + 10) (n + 11) 9 10 \ 305540235 n r 61108047 r | (n - 10) k + -------------------- - --------------------| (-252 2 ) /(n + 9)! 64 (n + 11) (n + 10) 64 (n + 11) (n + 10)/ / n | + 2 r! (n - r)! |- 12441 n (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) \ (n - 6) (n - 7) (n - 8) (n - 9) (n - 10) (n - 11) (n - 12) (n - 13)/(2048 12 11 10 9 (n + 15) (n + 14)) + 12441 n (105 n - 4095 n + 147875 n - 2277275 n 8 7 6 5 + 33270237 n - 253870617 n + 1876434417 n - 7307437137 n 4 3 2 + 28271361118 n - 51184761628 n + 96204101448 n - 57836888928 n 12 11 + 40495092480) r/(1024 (n + 15) (n + 14)) - 37323 (910 n - 20020 n 10 9 8 7 6 + 675675 n - 6831825 n + 93633540 n - 478227750 n + 3328249925 n 5 4 3 2 - 8247314075 n + 30178486338 n - 29673621978 n + 53306197308 n 2 - 9639481488 n + 6749182080) r /(512 (n + 15) (n + 14)) + 96231135 n ( 10 9 8 7 6 5 4 4 n - 55 n + 1782 n - 12012 n + 157872 n - 528528 n + 3534223 n 3 2 3 - 5226793 n + 18478339 n - 8200192 n + 14455920) r /(256 (n + 15) 10 9 8 7 6 (n + 14)) - 96231135 (198 n - 1782 n + 56232 n - 252252 n + 3223143 n 5 4 3 2 - 6819813 n + 44424897 n - 35458137 n + 122968538 n - 16400384 n 4 8 7 + 28911840) r /(1024 (n + 15) (n + 14)) + 12067384329 n (6 n - 36 n 6 5 4 3 2 + 1116 n - 3276 n + 41069 n - 51779 n + 331799 n - 119639 n + 411540) 5 8 7 6 r /(512 (n + 15) (n + 14)) - 4022461443 (90 n - 360 n + 11025 n 5 4 3 2 6 - 20475 n + 253449 n - 172809 n + 1097099 n - 119639 n + 411540) r /( 512 (n + 15) (n + 14)) + 8619560235 n 6 5 4 3 2 7 (72 n - 189 n + 5740 n - 6370 n + 78218 n - 24206 n + 153060) r /(256 (n + 15) (n + 14)) - 8619560235 6 5 4 3 2 8 (693 n - 1155 n + 34881 n - 21021 n + 256928 n - 24206 n + 153060) r / (1024 (n + 15) (n + 14)) 4 3 2 9 462583065945 n (11 n - 11 n + 331 n - 91 n + 1110) r + -------------------------------------------------------- 512 (n + 15) (n + 14) 4 3 2 10 92516613189 (66 n - 36 n + 1081 n - 91 n + 1110) r - ------------------------------------------------------- 512 (n + 15) (n + 14) 2 11 2 12 630795089925 n (4 n - n + 30) r 210265029975 (13 n - n + 30) r + ---------------------------------- - --------------------------------- 256 (n + 15) (n + 14) 512 (n + 15) (n + 14) 13 14 \ 436704293025 n r 62386327575 r | (n - 14) k + --------------------- - ---------------------| (-3432 2 ) / 256 (n + 15) (n + 14) 256 (n + 15) (n + 14)/ / n | (n + 13)! + 2 r! (n - r)! |- 449735 n (n - 1) (n - 2) (n - 3) (n - 4) \ (n - 5) (n - 6) (n - 7) (n - 8) (n - 9) (n - 10) (n - 11) (n - 12) (n - 13) (n - 14) (n - 15) (n - 16) (n - 17)/(65536 (n + 19) (n + 18)) + 25634895 n 16 15 14 13 12 11 (3 n - 204 n + 12716 n - 364140 n + 9791558 n - 152429004 n 10 9 8 7 + 2264838056 n - 20737062060 n + 183797955055 n - 999143042040 n 6 5 4 + 5342097893860 n - 16475066407800 n + 51143533311184 n 3 2 - 78728553219552 n + 127381990200768 n - 68681531577600 n 16 + 43619859763200) r/(32768 (n + 19) (n + 18)) - 25634895 (255 n 15 14 13 12 11 - 10200 n + 594150 n - 11834550 n + 298744264 n - 3389888502 n 10 9 8 7 + 47462936378 n - 317791818630 n + 2661258794337 n - 10276480840626 n 6 5 4 + 52028971928024 n - 105975701706420 n + 312551494925344 n 3 2 - 267973692419472 n + 415749480378048 n - 68681531577600 n 2 14 + 43619859763200) r /(32768 (n + 19) (n + 18)) + 2178966075 n (28 n 13 12 11 10 9 - 735 n + 41062 n - 584766 n + 14139580 n - 117162045 n 8 7 6 5 + 1572416131 n - 7586878728 n + 60986833622 n - 161690667138 n 4 3 2 + 787776405689 n - 987114134916 n + 2816911773328 n - 1113206701176 n 3 14 + 1696279353984) r /(8192 (n + 19) (n + 18)) - 2178966075 (2310 n 13 12 11 10 9 - 42042 n + 2284282 n - 23585562 n + 553159607 n - 3330618291 n 8 7 6 5 + 43346298996 n - 147519828456 n + 1151336447261 n - 2000942260317 n 4 3 2 + 9495344585272 n - 6580626366132 n + 18426002361456 n 4 - 2226413402352 n + 3392558707968) r /(32768 (n + 19) (n + 18)) + 12 11 10 9 8 2315369351295 n (14 n - 182 n + 9698 n - 72930 n + 1672957 n 7 6 5 4 3 - 7234656 n + 92060644 n - 214352320 n + 1638099229 n - 1746159012 n 2 5 + 8145600658 n - 2601015300 n + 7223138800) r /(16384 (n + 19) (n + 18)) 12 11 10 9 8 - 771789783765 (364 n - 3432 n + 180323 n - 984555 n + 22219197 n 7 6 5 4 - 67693626 n + 847365974 n - 1292180370 n + 9731931664 n 3 2 6 - 5740757217 n + 26513943378 n - 2601015300 n + 7223138800) r /(16384 10 9 8 (n + 19) (n + 18)) + 21499858262025 n (40 n - 275 n + 14300 n 7 6 5 4 3 - 56100 n + 1250844 n - 2608242 n + 32265158 n - 30212740 n 2 7 + 225374304 n - 61455357 n + 282589888) r /(8192 (n + 19) (n + 18)) - 10 9 8 7 6 21499858262025 (715 n - 3575 n + 184470 n - 510510 n + 11281725 n 5 4 3 2 - 15431325 n + 189318830 n - 98366590 n + 729643329 n - 61455357 n 8 8 + 282589888) r /(32768 (n + 19) (n + 18)) + 35833097103375 n (715 n 7 6 5 4 3 2 - 2574 n + 132054 n - 250614 n + 5502567 n - 4633146 n + 56530604 n 9 - 13630770 n + 100853805) r /(16384 (n + 19) (n + 18)) - 21499858262025 ( 8 7 6 5 4 3 3003 n - 7644 n + 390481 n - 487305 n + 10650682 n - 4991931 n 2 10 + 60709984 n - 4543590 n + 33617935) r /(16384 (n + 19) (n + 18)) + 396770111562825 n 6 5 4 3 2 11 (156 n - 273 n + 13902 n - 10710 n + 233368 n - 50949 n + 618716) r / (8192 (n + 19) (n + 18)) - 132256703854275 6 5 4 3 2 (1365 n - 1575 n + 80025 n - 34425 n + 748743 n - 50949 n + 618716) 12 r /(16384 (n + 19) (n + 18)) 4 3 2 13 4730720560941375 n (21 n - 15 n + 761 n - 153 n + 3325) r + -------------------------------------------------------------- 8192 (n + 19) (n + 18) 4 3 2 14 675817222991625 (120 n - 48 n + 2433 n - 153 n + 3325) r - ------------------------------------------------------------- 8192 (n + 19) (n + 18) 2 15 1486797890581575 n (16 n - 3 n + 152) r + ------------------------------------------ 4096 (n + 19) (n + 18) 2 16 17 1486797890581575 (51 n - 3 n + 152) r 9183163441827375 n r - ---------------------------------------- + ----------------------- 32768 (n + 19) (n + 18) 16384 (n + 19) (n + 18) 18 \ 1020351493536375 r | (n - 18) k - -----------------------| (-48620 2 ) /(n + 17)! 16384 (n + 19) (n + 18)/ and in Maple notation it is 2^n*r!*(n-r)!/(n+1)!*(2^n)^k+2^n*r!*(n-r)!/(n+5)!*(27/8*(n-1)*(n-2)*(n-3)*n-135 /4*n*(2*n^2-3*n+5)*r+(1215/4*n^2-405/4*n+675/4)*r^2-945/2*n*r^3+945/4*r^4)*(6*2 ^(n-4))^k+2^n*r!*(n-r)!/(n+9)!*(595/128*n*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*( n-7)-1785/32*n*(6*n^6-63*n^5+637*n^4-1960*n^3+6083*n^2-4949*n+4566)*r+(77381535 /64*n^2-6559875/16*n^3+4075155/16+187425/32*n^6-937125/32*n^5+17805375/64*n^4-\ 8833965/32*n)*r^2-687225/32*n*(2*n^4-5*n^3+46*n^2-28*n+81)*r^3+(10308375/64*n^4 -6185025/32*n^3+112017675/64*n^2-4810575/16*n+55665225/64)*r^4-5360355/32*n*(2* n^2-n+9)*r^5+(12507495/32*n^2-1786785/32*n+16081065/32)*r^6-3828825/16*n*r^7+ 3828825/64*r^8)*(70*2^(n-8))^k+2^n*r!*(n-r)!/(n+13)!*(5775/1024*n*(n-1)*(n-2)*( n-3)*(n-4)*(n-5)*(n-6)*(n-7)*(n-8)*(n-9)*(n-10)*(n-11)-75075/512*n*(6*n^10-165* n^9+4235*n^8-43560*n^7+429132*n^6-2004849*n^5+9223335*n^4-18770730*n^3+39343612 *n^2-25465176*n+19056960)*r+(17342325/512*n^10+22354707375/8+2490479266275/128* n^2-1245562943625/128*n^3+370380035025/512*n^6-1061610424875/512*n^5+ 2303519092875/256*n^4-238974761025/64*n-20030385375/256*n^7+780404625/64*n^8-\ 260134875/512*n^9)*r^2-5780775/256*n*(25*n^8-225*n^7+5190*n^6-20790*n^5+184701* n^4-307395*n^3+1288240*n^2-618090*n+1212328)*r^3+(25109171962875/512*n^2-\ 3053203027875/256*n^3+876024424275/64+81942485625/128*n^6-100151926875/64*n^5+ 6941699139375/512*n^4-1786519609875/256*n-1820944125/64*n^7+1300674375/256*n^8) *r^4-294819525/256*n*(24*n^6-84*n^5+1860*n^4-2640*n^3+22526*n^2-8811*n+35945)*r ^5+(3532429275375/256+926028128025/32*n^2-437806994625/128*n^3+6191210025/64*n^ 6-13266878625/64*n^5+1163063026125/256*n^4-865884944925/256*n)*r^6-800224425/ 128*n*(36*n^4-45*n^3+980*n^2-330*n+2782)*r^7+(180050495625/512*n^4-60016831875/ 256*n^3+2604730503375/512*n^2-132037030125/256*n+1113112175175/256)*r^8-\ 9335951625/256*n*(10*n^2-3*n+65)*r^9+(61617280725/256*n^2-5601570975/256*n+ 121367371125/256)*r^10-11712375675/128*n*r^11+3904125225/256*r^12)*(924*2^(n-12 ))^k+2^n*r!*(n-r)!/(n+17)!*(212355/32768*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*(n -7)*(n-8)*(n-9)*(n-10)*(n-11)*(n-12)*(n-13)*(n-14)*(n-15)*n-3610035/4096*n*(2*n ^14-105*n^13+5075*n^12-109200*n^11+2215577*n^10-24954930*n^9+269790235*n^8-\ 1691718600*n^7+10363418065*n^6-35376706365*n^5+120835413790*n^4-199949349600*n^ 3+345786211656*n^2-195883887600*n+130026254400)*r+(371508005113245/8192*n^10-\ 14783093325/4096*n^13+1026716033800758015/1024*n^2+1335406097025/8192*n^12+ 7334364520357875/64-2439210398625/1024*n^11-310050768641488875/512*n^3+ 487354725/4096*n^14+756310216422946185/8192*n^6-839901935101954725/4096*n^5+ 2719847483007229575/4096*n^4-44196730635754125/256*n-16340270460388875/1024*n^7 +30018015002514825/8192*n^8-737129382464475/2048*n^9)*r^2-342953325/4096*n*(42* n^12-819*n^11+35490*n^10-360360*n^9+6573567*n^8-36540504*n^7+356791292*n^6-\ 1047506460*n^5+5826890069*n^4-7862872017*n^3+24617748428*n^2-10255518000*n+ 16670279016)*r^3+(260561875248675/8192*n^10+13910078982259134675/4096*n^2+ 468131288625/8192*n^12-3089666504925/4096*n^11-4528840372140909825/4096*n^3+ 714640952151866025/1024+298392095231255025/2048*n^6-282925911300157125/1024*n^5 +24523650801798323175/16384*n^4-219822749856084375/512*n-31468253352661125/2048 *n^7+65652323563151325/16384*n^8-231724987869375/1024*n^9)*r^4-18725251545/4096 *n*(126*n^10-1155*n^9+47795*n^8-237600*n^7+4119698*n^6-10619070*n^5+98656580*n^ 4-113384700*n^3+603937851*n^2-203823675*n+620896950)*r^5+(15860288058615/4096*n ^10+1937741928712214625/2048+24708604555434654855/8192*n^2-585538147680861375/ 1024*n^3+243820208325088395/4096*n^6-202932385709978925/2048*n^5+ 3718552673669959125/4096*n^4-1272216528400442625/4096*n-3568564813188375/1024*n ^7+8326651230772875/8192*n^8-101958994662525/4096*n^9)*r^6-3383920457775/4096*n *(22*n^8-99*n^7+4004*n^6-9240*n^5+156170*n^4-157836*n^3+1433116*n^2-414000*n+ 2174487)*r^7+(7358291044465786425/16384+7899873978050844075/8192*n^2-\ 218658788220047175/2048*n^3+62795411934930675/8192*n^6-11725284386190375/1024*n ^5+3147800639992833825/16384*n^4-87558941844928125/1024*n-781685625746025/4096* n^7+1005024375959175/16384*n^8)*r^8-5639867429625/4096*n*(110*n^6-231*n^5+9233* n^4-8400*n^3+140273*n^2-35910*n+323595)*r^9+(365006580177900375/4096+ 1025579436793186275/8192*n^2-7698419041438125/1024*n^3+1129101459410925/4096*n^ 6-1539683808287625/4096*n^5+122661476726914125/8192*n^4-20252763939783375/2048* n)*r^10-6460211783025/4096*n*(234*n^4-195*n^3+7750*n^2-1800*n+29937)*r^11+( 2939396361276375/8192*n^4-678322237217625/4096*n^3+53845865211513375/8192*n^2-\ 484515883726875/1024*n+64466453382806475/8192)*r^12-72056208349125/4096*n*(14*n ^2-3*n+119)*r^13+(463218482244375/4096*n^2-30881232149625/4096*n+ 1224955541935125/4096)*r^14-63821213109225/2048*n*r^15+63821213109225/16384*r^ 16)*(12870*2^(n-16))^k+2^n*r!*(n-r)!/(n+1)!*(-5/2*n*(n-1)/(n+3)/(n+2)+15*n/(n+3 )/(n+2)*r-15/(n+3)/(n+2)*r^2)*(-2*2^(n-2))^k+2^n*r!*(n-r)!/(n+5)!*(-65/16*n*(n-\ 1)*(n-2)*(n-3)*(n-4)*(n-5)/(n+7)/(n+6)+1365/8*n*(n^4-5*n^3+25*n^2-25*n+28)/(n+7 )/(n+6)*r-1365/8*(10*n^4-20*n^3+95*n^2-25*n+28)/(n+7)/(n+6)*r^2+6825/4*n*(4*n^2 -3*n+14)/(n+7)/(n+6)*r^3-6825/8*(15*n^2-3*n+14)/(n+7)/(n+6)*r^4+45045/4*n/(n+7) /(n+6)*r^5-15015/4/(n+7)/(n+6)*r^6)*(-20*2^(n-6))^k+2^n*r!*(n-r)!/(n+9)!*(-1323 /256*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*(n-7)*(n-8)*(n-9)*n/(n+11)/(n+10)+ 14553/128*n*(5*n^8-90*n^7+1530*n^6-9450*n^5+56847*n^4-136080*n^3+332290*n^2-\ 236700*n+193248)/(n+10)/(n+11)*r-43659/128*(45*n^8-420*n^7+6685*n^6-23625*n^5+ 133700*n^4-164115*n^3+381510*n^2-78900*n+64416)/(n+10)/(n+11)*r^2+945945/64*n*( 12*n^6-63*n^5+966*n^4-1890*n^3+10332*n^2-5481*n+12496)/(n+10)/(n+11)*r^3-945945 /128*(147*n^6-441*n^5+6615*n^4-6615*n^3+35511*n^2-5481*n+12496)/(n+10)/(n+11)*r ^4+3972969/64*n*(63*n^4-105*n^3+1555*n^2-675*n+3597)/(n+10)/(n+11)*r^5-3972969/ 64*(140*n^4-120*n^3+1765*n^2-225*n+1199)/(n+10)/(n+11)*r^6+48243195/32*n*(8*n^2 -3*n+44)/(n+10)/(n+11)*r^7-48243195/128*(27*n^2-3*n+44)/(n+10)/(n+11)*r^8+ 305540235/64*n/(n+11)/(n+10)*r^9-61108047/64/(n+11)/(n+10)*r^10)*(-252*2^(n-10) )^k+2^n*r!*(n-r)!/(n+13)!*(-12441/2048*n*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*(n -7)*(n-8)*(n-9)*(n-10)*(n-11)*(n-12)*(n-13)/(n+15)/(n+14)+12441/1024*n*(105*n^ 12-4095*n^11+147875*n^10-2277275*n^9+33270237*n^8-253870617*n^7+1876434417*n^6-\ 7307437137*n^5+28271361118*n^4-51184761628*n^3+96204101448*n^2-57836888928*n+ 40495092480)/(n+15)/(n+14)*r-37323/512*(910*n^12-20020*n^11+675675*n^10-6831825 *n^9+93633540*n^8-478227750*n^7+3328249925*n^6-8247314075*n^5+30178486338*n^4-\ 29673621978*n^3+53306197308*n^2-9639481488*n+6749182080)/(n+15)/(n+14)*r^2+ 96231135/256*n*(4*n^10-55*n^9+1782*n^8-12012*n^7+157872*n^6-528528*n^5+3534223* n^4-5226793*n^3+18478339*n^2-8200192*n+14455920)/(n+15)/(n+14)*r^3-96231135/ 1024*(198*n^10-1782*n^9+56232*n^8-252252*n^7+3223143*n^6-6819813*n^5+44424897*n ^4-35458137*n^3+122968538*n^2-16400384*n+28911840)/(n+15)/(n+14)*r^4+ 12067384329/512*n*(6*n^8-36*n^7+1116*n^6-3276*n^5+41069*n^4-51779*n^3+331799*n^ 2-119639*n+411540)/(n+15)/(n+14)*r^5-4022461443/512*(90*n^8-360*n^7+11025*n^6-\ 20475*n^5+253449*n^4-172809*n^3+1097099*n^2-119639*n+411540)/(n+15)/(n+14)*r^6+ 8619560235/256*n*(72*n^6-189*n^5+5740*n^4-6370*n^3+78218*n^2-24206*n+153060)/(n +15)/(n+14)*r^7-8619560235/1024*(693*n^6-1155*n^5+34881*n^4-21021*n^3+256928*n^ 2-24206*n+153060)/(n+15)/(n+14)*r^8+462583065945/512*n*(11*n^4-11*n^3+331*n^2-\ 91*n+1110)/(n+15)/(n+14)*r^9-92516613189/512*(66*n^4-36*n^3+1081*n^2-91*n+1110) /(n+15)/(n+14)*r^10+630795089925/256*n*(4*n^2-n+30)/(n+15)/(n+14)*r^11-\ 210265029975/512*(13*n^2-n+30)/(n+15)/(n+14)*r^12+436704293025/256*n/(n+15)/(n+ 14)*r^13-62386327575/256/(n+15)/(n+14)*r^14)*(-3432*2^(n-14))^k+2^n*r!*(n-r)!/( n+17)!*(-449735/65536*n*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*(n-7)*(n-8)*(n-9)*( n-10)*(n-11)*(n-12)*(n-13)*(n-14)*(n-15)*(n-16)*(n-17)/(n+19)/(n+18)+25634895/ 32768*n*(3*n^16-204*n^15+12716*n^14-364140*n^13+9791558*n^12-152429004*n^11+ 2264838056*n^10-20737062060*n^9+183797955055*n^8-999143042040*n^7+5342097893860 *n^6-16475066407800*n^5+51143533311184*n^4-78728553219552*n^3+127381990200768*n ^2-68681531577600*n+43619859763200)/(n+19)/(n+18)*r-25634895/32768*(255*n^16-\ 10200*n^15+594150*n^14-11834550*n^13+298744264*n^12-3389888502*n^11+47462936378 *n^10-317791818630*n^9+2661258794337*n^8-10276480840626*n^7+52028971928024*n^6-\ 105975701706420*n^5+312551494925344*n^4-267973692419472*n^3+415749480378048*n^2 -68681531577600*n+43619859763200)/(n+19)/(n+18)*r^2+2178966075/8192*n*(28*n^14-\ 735*n^13+41062*n^12-584766*n^11+14139580*n^10-117162045*n^9+1572416131*n^8-\ 7586878728*n^7+60986833622*n^6-161690667138*n^5+787776405689*n^4-987114134916*n ^3+2816911773328*n^2-1113206701176*n+1696279353984)/(n+19)/(n+18)*r^3-\ 2178966075/32768*(2310*n^14-42042*n^13+2284282*n^12-23585562*n^11+553159607*n^ 10-3330618291*n^9+43346298996*n^8-147519828456*n^7+1151336447261*n^6-\ 2000942260317*n^5+9495344585272*n^4-6580626366132*n^3+18426002361456*n^2-\ 2226413402352*n+3392558707968)/(n+19)/(n+18)*r^4+2315369351295/16384*n*(14*n^12 -182*n^11+9698*n^10-72930*n^9+1672957*n^8-7234656*n^7+92060644*n^6-214352320*n^ 5+1638099229*n^4-1746159012*n^3+8145600658*n^2-2601015300*n+7223138800)/(n+19)/ (n+18)*r^5-771789783765/16384*(364*n^12-3432*n^11+180323*n^10-984555*n^9+ 22219197*n^8-67693626*n^7+847365974*n^6-1292180370*n^5+9731931664*n^4-\ 5740757217*n^3+26513943378*n^2-2601015300*n+7223138800)/(n+19)/(n+18)*r^6+ 21499858262025/8192*n*(40*n^10-275*n^9+14300*n^8-56100*n^7+1250844*n^6-2608242* n^5+32265158*n^4-30212740*n^3+225374304*n^2-61455357*n+282589888)/(n+19)/(n+18) *r^7-21499858262025/32768*(715*n^10-3575*n^9+184470*n^8-510510*n^7+11281725*n^6 -15431325*n^5+189318830*n^4-98366590*n^3+729643329*n^2-61455357*n+282589888)/(n +19)/(n+18)*r^8+35833097103375/16384*n*(715*n^8-2574*n^7+132054*n^6-250614*n^5+ 5502567*n^4-4633146*n^3+56530604*n^2-13630770*n+100853805)/(n+19)/(n+18)*r^9-\ 21499858262025/16384*(3003*n^8-7644*n^7+390481*n^6-487305*n^5+10650682*n^4-\ 4991931*n^3+60709984*n^2-4543590*n+33617935)/(n+19)/(n+18)*r^10+396770111562825 /8192*n*(156*n^6-273*n^5+13902*n^4-10710*n^3+233368*n^2-50949*n+618716)/(n+19)/ (n+18)*r^11-132256703854275/16384*(1365*n^6-1575*n^5+80025*n^4-34425*n^3+748743 *n^2-50949*n+618716)/(n+19)/(n+18)*r^12+4730720560941375/8192*n*(21*n^4-15*n^3+ 761*n^2-153*n+3325)/(n+19)/(n+18)*r^13-675817222991625/8192*(120*n^4-48*n^3+ 2433*n^2-153*n+3325)/(n+19)/(n+18)*r^14+1486797890581575/4096*n*(16*n^2-3*n+152 )/(n+19)/(n+18)*r^15-1486797890581575/32768*(51*n^2-3*n+152)/(n+19)/(n+18)*r^16 +9183163441827375/16384*n/(n+19)/(n+18)*r^17-1020351493536375/16384/(n+19)/(n+ 18)*r^18)*(-48620*2^(n-18))^k In particular, for our object of interest, when r=0, i.e. 1 / | n | (P[k](exp(2 I Pi t)) P[k](exp(-2 I Pi t))) dt | / 0 n n k n (n - 4) k 2 (2 ) 27 2 (n - 1) (n - 2) (n - 3) n (6 2 ) n gu := -------- + --------------------------------------------- + 595 2 n n + 1 8 (n + 1) (n + 2) (n + 3) (n + 4) (n + 5) (n - 8) k (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (n - 7) (70 2 ) /(128 (n + 1) (n + 2) (n + 3) (n + 4) (n + 5) (n + 6) (n + 7) (n + 8) (n + 9)) + n 5775 2 n (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (n - 7) (n - 8) (n - 12) k (n - 9) (n - 10) (n - 11) (924 2 ) /(1024 (n + 1) (n + 2) (n + 3) (n + 4) (n + 5) (n + 6) (n + 7) (n + 8) (n + 9) (n + 10) (n + 11) (n + 12) n (n + 13)) + 212355 2 (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (n - 7) (n - 8) (n - 9) (n - 10) (n - 11) (n - 12) (n - 13) (n - 14) (n - 16) k (n - 15) n (12870 2 ) /(32768 (n + 1) (n + 2) (n + 3) (n + 4) (n + 5) (n + 6) (n + 7) (n + 8) (n + 9) (n + 10) (n + 11) (n + 12) (n + 13) (n - 1) (n - 2) k 5 2 n (n - 1) (-2 2 ) (n + 14) (n + 15) (n + 16) (n + 17)) - ----------------------------------- (n + 1) (n + 3) (n + 2) n (n - 6) k 65 2 n (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (-20 2 ) - --------------------------------------------------------------- - 1323 16 (n + 1) (n + 2) (n + 3) (n + 4) (n + 5) (n + 7) (n + 6) n 2 (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (n - 7) (n - 8) (n - 9) (n - 10) k n (-252 2 ) /(256 (n + 1) (n + 2) (n + 3) (n + 4) (n + 5) (n + 6) n (n + 7) (n + 8) (n + 9) (n + 11) (n + 10)) - 12441 2 n (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (n - 7) (n - 8) (n - 9) (n - 10) (n - 11) (n - 14) k (n - 12) (n - 13) (-3432 2 ) /(2048 (n + 1) (n + 2) (n + 3) (n + 4) (n + 5) (n + 6) (n + 7) (n + 8) (n + 9) (n + 10) (n + 11) (n + 12) (n + 13) n (n + 15) (n + 14)) - 449735 2 n (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (n - 7) (n - 8) (n - 9) (n - 10) (n - 11) (n - 12) (n - 13) (n - 18) k (n - 14) (n - 15) (n - 16) (n - 17) (-48620 2 ) /(65536 (n + 1) (n + 2) (n + 3) (n + 4) (n + 5) (n + 6) (n + 7) (n + 8) (n + 9) (n + 10) (n + 11) (n + 12) (n + 13) (n + 14) (n + 15) (n + 16) (n + 17) (n + 19) (n + 18)) n n k n (n - 4) k 2 (2 ) 27 2 (n - 1) (n - 2) (n - 3) n (6 2 ) n -------- + --------------------------------------------- + 595 2 n (n - 1) n + 1 8 (n + 1) (n + 2) (n + 3) (n + 4) (n + 5) (n - 8) k (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (n - 7) (70 2 ) /(128 (n + 1) n (n + 2) (n + 3) (n + 4) (n + 5) (n + 6) (n + 7) (n + 8) (n + 9)) + 5775 2 n (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (n - 7) (n - 8) (n - 9) (n - 12) k (n - 10) (n - 11) (924 2 ) /(1024 (n + 1) (n + 2) (n + 3) (n + 4) (n + 5) (n + 6) (n + 7) (n + 8) (n + 9) (n + 10) (n + 11) (n + 12) (n + 13) n ) + 212355 2 (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (n - 7) (n - 8) (n - 9) (n - 10) (n - 11) (n - 12) (n - 13) (n - 14) (n - 15) n (n - 16) k (12870 2 ) /(32768 (n + 1) (n + 2) (n + 3) (n + 4) (n + 5) (n + 6) (n + 7) (n + 8) (n + 9) (n + 10) (n + 11) (n + 12) (n + 13) (n + 14) (n - 1) (n - 2) k 5 2 n (n - 1) (-2 2 ) (n + 15) (n + 16) (n + 17)) - ----------------------------------- (n + 1) (n + 3) (n + 2) n (n - 6) k 65 2 n (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (-20 2 ) - --------------------------------------------------------------- - 1323 16 (n + 1) (n + 2) (n + 3) (n + 4) (n + 5) (n + 7) (n + 6) n 2 (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (n - 7) (n - 8) (n - 9) (n - 10) k n (-252 2 ) /(256 (n + 1) (n + 2) (n + 3) (n + 4) (n + 5) (n + 6) n (n + 7) (n + 8) (n + 9) (n + 11) (n + 10)) - 12441 2 n (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (n - 7) (n - 8) (n - 9) (n - 10) (n - 11) (n - 14) k (n - 12) (n - 13) (-3432 2 ) /(2048 (n + 1) (n + 2) (n + 3) (n + 4) (n + 5) (n + 6) (n + 7) (n + 8) (n + 9) (n + 10) (n + 11) (n + 12) (n + 13) n (n + 15) (n + 14)) - 449735 2 n (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (n - 7) (n - 8) (n - 9) (n - 10) (n - 11) (n - 12) (n - 13) (n - 18) k (n - 14) (n - 15) (n - 16) (n - 17) (-48620 2 ) /(65536 (n + 1) (n + 2) (n + 3) (n + 4) (n + 5) (n + 6) (n + 7) (n + 8) (n + 9) (n + 10) (n + 11) (n + 12) (n + 13) (n + 14) (n + 15) (n + 16) (n + 17) (n + 19) (n + 18)) and in Maple notation it is 2^n/(n+1)*(2^n)^k+27/8*2^n/(n+1)/(n+2)/(n+3)/(n+4)/(n+5)*(n-1)*(n-2)*(n-3)*n*(6 *2^(n-4))^k+595/128*2^n/(n+1)/(n+2)/(n+3)/(n+4)/(n+5)/(n+6)/(n+7)/(n+8)/(n+9)*n *(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*(n-7)*(70*2^(n-8))^k+5775/1024*2^n/(n+1)/( n+2)/(n+3)/(n+4)/(n+5)/(n+6)/(n+7)/(n+8)/(n+9)/(n+10)/(n+11)/(n+12)/(n+13)*n*(n -1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*(n-7)*(n-8)*(n-9)*(n-10)*(n-11)*(924*2^(n-12) )^k+212355/32768*2^n/(n+1)/(n+2)/(n+3)/(n+4)/(n+5)/(n+6)/(n+7)/(n+8)/(n+9)/(n+ 10)/(n+11)/(n+12)/(n+13)/(n+14)/(n+15)/(n+16)/(n+17)*(n-1)*(n-2)*(n-3)*(n-4)*(n -5)*(n-6)*(n-7)*(n-8)*(n-9)*(n-10)*(n-11)*(n-12)*(n-13)*(n-14)*(n-15)*n*(12870* 2^(n-16))^k-5*2^(n-1)/(n+1)*n*(n-1)/(n+3)/(n+2)*(-2*2^(n-2))^k-65/16*2^n/(n+1)/ (n+2)/(n+3)/(n+4)/(n+5)*n*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)/(n+7)/(n+6)*(-20*2^(n-6 ))^k-1323/256*2^n/(n+1)/(n+2)/(n+3)/(n+4)/(n+5)/(n+6)/(n+7)/(n+8)/(n+9)*(n-1)*( n-2)*(n-3)*(n-4)*(n-5)*(n-6)*(n-7)*(n-8)*(n-9)*n/(n+11)/(n+10)*(-252*2^(n-10))^ k-12441/2048*2^n/(n+1)/(n+2)/(n+3)/(n+4)/(n+5)/(n+6)/(n+7)/(n+8)/(n+9)/(n+10)/( n+11)/(n+12)/(n+13)*n*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*(n-7)*(n-8)*(n-9)*(n-\ 10)*(n-11)*(n-12)*(n-13)/(n+15)/(n+14)*(-3432*2^(n-14))^k-449735/65536*2^n/(n+1 )/(n+2)/(n+3)/(n+4)/(n+5)/(n+6)/(n+7)/(n+8)/(n+9)/(n+10)/(n+11)/(n+12)/(n+13)/( n+14)/(n+15)/(n+16)/(n+17)*n*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*(n-7)*(n-8)*(n -9)*(n-10)*(n-11)*(n-12)*(n-13)*(n-14)*(n-15)*(n-16)*(n-17)/(n+19)/(n+18)*(-\ 48620*2^(n-18))^k --------------------------------- This took, 236.313, seconds.