If the kernel is, binomial(n, k), then the MWZ pair is the following consisting of a matrix of dimension, 1 k [binomial(n, k), [[1/2]], [1], [- 1/2 ---------]] n + 1 - k 2 If the kernel is, binomial(n, k) , then the MWZ pair is the following consisting of a matrix of dimension, 1 2 2 n + 1 k (3 n + 3 - 2 k) [binomial(n, k) , [[1/2 -------]], [1], [- 1/2 ----------------------]] 2 n + 1 2 (2 n + 1) (n + 1 - k) 3 If the kernel is, binomial(n, k) , then the MWZ pair is the following consisting of a matrix of dimension, 3 3 [binomial(n, k) , [[1/2, 3/4 n + 1/2, 3/8 n (n + 1)], 1 7 n + 4 [- 3/2 -----, - 1/4 -------, - 3/8 n + 1/4], n + 1 n + 1 1 n 5 n + 2 2 [3/2 --------, 3/4 --------, - 1/8 -------]], [1, k, k ], [ 2 2 n + 1 (n + 1) (n + 1) 3 3 k k (8 + 6 n - 4 k) - 1/2 ------------, 1/8 ------------------, 3 3 (n + 1 - k) (n + 1 - k) 3 2 2 k (3 n - 6 - 3 n + 10 k + 6 n k - 4 k ) 1/8 -----------------------------------------]] 3 (n + 1 - k) 4 If the kernel is, binomial(n, k) , then the MWZ pair is the following consisting of a matrix of dimension, 3 2 4 n + 1 (n + 1) (10 n + 10 n + 3) [binomial(n, k) , [[3/2 -------, 1/4 --------------------------, 4 n + 3 (2 n + 1) (4 n + 3) 3 2 n (n + 1) (20 n + 30 n + 15 n + 2) 1/4 ------------------------------------], [ (4 n + 1) (2 n + 1) (4 n + 3) 2 4 3 2 5 14 n + 12 n + 3 20 n + 20 n + 2 n - 4 n - 1 - -------, - 1/2 -------------------, - 1/2 ------------------------------] 4 n + 3 (2 n + 1) (4 n + 3) (4 n + 1) (2 n + 1) (4 n + 3) , 3 2 5 n 2 n - 7 n - 5 n - 1 [-----------------, 5/2 -----------------, 1/2 ---------------------------] (n + 1) (4 n + 3) (n + 1) (4 n + 3) (4 n + 1) (4 n + 3) (n + 1) 2 ], [1, k, k ], [ 4 2 3 2 k (32 n k + 24 n k - 70 n + 4 k - 80 n - 140 n - 10) 4 1/4 --------------------------------------------------------, 1/4 k ( 4 (4 n + 1) (2 n + 1) (4 n + 3) (n + 1 - k) 2 3 2 2 2 2 4 -112 n k - 180 n k - 80 k n + 32 k n + 32 k n + 6 k + 40 n + 105 n 2 3 / + 220 n + 170 n - 18 k + 15) / ((4 n + 1) (2 n + 1) (4 n + 3) / 4 4 4 2 3 2 5 (n + 1 - k) ), 1/4 k (10 n - 110 n - 195 n - 115 n - 42 k + 20 n 3 2 3 2 2 2 3 2 3 + 50 k + 12 k - 80 k n - 176 k n - 220 k n + 32 k n + 40 k n 3 2 4 / + 248 n k + 240 k n + 400 n k + 40 k n - 20) / ((4 n + 1) (2 n + 1) / 4 (4 n + 3) (n + 1 - k) )]] 5 If the kernel is, binomial(n, k) , then the MWZ pair is the following consisting of a matrix of dimension, 5 5 [binomial(n, k) , [[1/2, 5/4 n + 1, 5/8 n (n + 1), 35 3 65 2 2 - -- n - -- n - 5/2 n - 1/2, - 5/32 n (n + 1) (23 n + 23 n + 6)], [ 16 16 3 2 1 23 n + 18 37 n + 69 n + 42 n + 8 - 5/2 -----, - 1/4 ---------, - 15/8 n + 3/4, 5/16 ------------------------, n + 1 n + 1 n + 1 505 3 495 2 5 4 n + 3 2 n - 13 --- n + --- n + 15/4 n - 1/8], [--------, 5/2 --------, 1/4 --------, 32 32 2 2 n + 1 (n + 1) (n + 1) 3 2 3 2 190 n + 351 n + 207 n + 36 78 n + 71 n + 13 n - 2 - 1/8 ----------------------------, - 5/16 ------------------------], [ 2 n + 1 (n + 1) 3 2 5 3 n + 2 3 n + 4 179 n + 317 n + 172 n + 24 - --------, - 5/2 --------, 5/4 --------, 1/8 ----------------------------, 3 3 2 3 (n + 1) (n + 1) (n + 1) (n + 1) 2 (15 n + 8) (13 n + n - 2) 1 n 1/16 --------------------------], [5/2 --------, 5/4 --------, 2 4 4 (n + 1) (n + 1) (n + 1) 2 7 n + 4 n (23 n + 33 n + 12) - 5/8 --------, - 5/16 ---------------------, 3 4 (n + 1) (n + 1) 3 2 141 n + 233 n + 118 n + 16 2 3 4 1/32 ----------------------------]], [1, k, k , k , k ], [ 3 (n + 1) 5 5 k k (-48 - 40 n + 16 k) - 1/2 ------------, - 1/32 ----------------------, 5 5 (n + 1 - k) (n + 1 - k) 5 2 2 k (-40 n k + 40 - 56 k - 20 n + 20 n + 16 k ) 5 2 - 1/32 -----------------------------------------------, - 1/32 k (-40 k n 5 (n + 1 - k) 2 2 2 3 3 + 60 n k - 20 n k - 16 + 80 k - 64 k + 150 n + 60 n + 70 n + 16 k ) / 5 5 3 2 2 2 / (n + 1 - k) , - 1/32 k (-72 k + 100 k n - 20 k n + 20 - 84 k / 2 2 4 3 4 3 2 + 120 k + 45 n + 20 n + 115 n + 160 n + 16 k + 70 k n + 120 n k 3 / 5 - 50 n k - 40 k n) / (n + 1 - k) ]] / 6 If the kernel is, binomial(n, k) , then the MWZ pair is the following consisting of a matrix of dimension, 5 2 6 n + 1 (n + 1) (21 n + 28 n + 10) [binomial(n, k) , [[5/2 -------, 5/12 ---------------------------, 6 n + 5 (3 n + 2) (6 n + 5) 3 2 (n + 1) (21 n + 42 n + 28 n + 6) n 5/12 ------------------------------------, (2 n + 1) (3 n + 2) (6 n + 5) 5 4 3 2 2 (357 n + 1043 n + 1204 n + 672 n + 184 n + 20) (n + 1) - 1/24 -----------------------------------------------------------, - 1/24 (6 n + 5) (3 n + 2) (2 n + 1) (3 n + 1) n (n + 1) ( 7 6 5 4 3 2 5103 n + 19026 n + 29946 n + 25690 n + 12915 n + 3778 n + 586 n + 36) 14 /((6 n + 5) (3 n + 2) (2 n + 1) (3 n + 1) (6 n + 1))], [- -------, 6 n + 5 2 4 3 2 45 n + 58 n + 20 105 n + 175 n + 72 n - 15 n - 10 - -------------------, - 1/3 -----------------------------------, (3 n + 2) (6 n + 5) (2 n + 1) (3 n + 2) (6 n + 5) 4 3 2 (11 n + 5) (n + 1) (21 n + 49 n + 42 n + 15 n + 2) 1/2 -----------------------------------------------------, 1/6 (21 n (6 n + 5) (3 n + 2) (2 n + 1) (3 n + 1) 6 5 3 2 4 8 + 41146 n - 2 + 34713 n + 4699 n + 642 n + 16953 n + 7119 n 7 + 26397 n )/((6 n + 5) (3 n + 2) (2 n + 1) (3 n + 1) (6 n + 1))], [ 2 1 51 n + 62 n + 20 63/2 -----------------, 7/4 ---------------------------, (n + 1) (6 n + 5) (6 n + 5) (3 n + 2) (n + 1) 4 3 2 165 n + 112 n - 198 n - 226 n - 60 1/4 -------------------------------------, (6 n + 5) (3 n + 2) (2 n + 1) (n + 1) 5 4 3 2 2625 n + 6965 n + 7228 n + 3594 n + 856 n + 80 8 - 1/8 --------------------------------------------------, - 1/8 (19971 n (6 n + 5) (3 n + 2) (2 n + 1) (3 n + 1) 5 6 3 2 4 + 87330 n - 104 n + 108976 n + 9164 n + 700 n + 39167 n - 16 7 + 72324 n )/((6 n + 5) (3 n + 2) (2 n + 1) (3 n + 1) (6 n + 1) (n + 1))], 2 35 69 n + 74 n + 20 [- ------------------, - 7/6 ----------------------------, 2 2 (6 n + 5) (n + 1) (3 n + 2) (6 n + 5) (n + 1) 4 3 2 3 n + 84 n + 150 n + 94 n + 20 7/6 --------------------------------------, 2 (2 n + 1) (3 n + 2) (6 n + 5) (n + 1) 5 4 3 2 5055 n + 12527 n + 11980 n + 5370 n + 1096 n + 80 1/12 -----------------------------------------------------, 1/12 (-40 (6 n + 5) (3 n + 2) (2 n + 1) (3 n + 1) (n + 1) 3 6 5 4 2 7 + 2000 n + 128376 n - 476 n + 88922 n + 29729 n - 1644 n + 93240 n 8 / + 27405 n ) / ((6 n + 5) (3 n + 2) (2 n + 1) (3 n + 1) (6 n + 1) / 2 1 n (n + 1) )], [35/2 ------------------, 35/4 ------------------, 3 3 (6 n + 5) (n + 1) (6 n + 5) (n + 1) 3 2 2 17 n + 54 n + 42 n + 10 (27 n + 32 n + 10) n - 7/12 ----------------------------, - 7/8 ----------------------------, 3 2 (6 n + 5) (2 n + 1) (n + 1) (6 n + 5) (2 n + 1) (n + 1) 6 5 4 3 2 1765 n + 3027 n + 843 n - 1223 n - 942 n - 228 n - 20 - 1/24 ----------------------------------------------------------]], 3 (6 n + 5) (2 n + 1) (6 n + 1) (n + 1) 2 3 4 6 4 2 3 [1, k, k , k , k ], [- 1/24 k (-2592 k n - 2520 n k - 4320 k n 2 4 3 5 - 600 n k + 10920 n + 24192 n + 23940 n + 9072 n + 2268 n - 48 k + 168 / 6 ) / ((6 n + 1) (3 n + 1) (2 n + 1) (3 n + 2) (6 n + 5) (n + 1 - k) ), - / 6 2 3 2 3 5 2 4 1/24 k (-280 - 60 k + 29556 k n - 4752 k n - 34272 n - 2592 k n 6 2 2 2 2 4 + 3144 n k - 7560 n - 2952 k n - 732 k n + 14436 n k + 27216 k n 5 3 4 2 / + 9072 k n - 47684 n - 58254 n - 19754 n - 3892 n + 240 k) / ( / 6 6 (6 n + 1) (3 n + 1) (2 n + 1) (3 n + 2) (6 n + 5) (n + 1 - k) ), - 1/24 k 2 4 3 3 2 (280 + 3780 n - 34992 n k - 560 k - 83076 k n - 936 k n - 3528 k n 2 3 2 2 2 3 3 3 + 36252 k n + 19356 k n + 4512 k n - 76648 k n - 80 k + 40740 n 5 4 6 7 5 2 + 14532 n + 42378 n - 5670 n - 3780 n - 42084 k n + 18284 n 2 2 4 6 3 4 3 3 - 7416 n k + 360 k + 30240 k n - 7560 k n - 2592 k n - 5184 k n 2 5 / + 9072 k n ) / ((6 n + 1) (3 n + 1) (2 n + 1) (3 n + 2) (6 n + 5) / 6 6 2 4 (n + 1 - k) ), - 1/24 k (-280 - 3556 n + 54732 n k + 960 k + 120330 k n 3 3 2 2 3 2 2 2 + 6900 k n + 26256 k n - 114960 k n - 58698 k n - 13908 k n 3 3 3 5 4 6 + 115404 k n + 600 k - 29764 n + 15113 n - 19061 n + 29309 n 7 5 2 2 2 4 + 14679 n + 54096 k n - 15638 n + 12192 n k - 1140 k - 111426 k n 6 3 4 3 3 2 5 8 4 + 1890 k n + 33264 k n + 44028 k n - 49896 k n + 2142 n - 120 k 4 4 3 3 5 7 2 6 - 1284 k n - 5616 k n + 9072 k n - 3780 k n - 7560 k n 4 4 4 2 / - 2592 k n - 4248 k n ) / ((6 n + 1) (3 n + 1) (2 n + 1) (3 n + 2) / 6 6 4 5 2 (6 n + 5) (n + 1 - k) ), - 1/24 k (392 + 4284 n + 9072 k n - 75360 n k 5 2 5 4 2 7 3 6 5 - 1792 k - 5112 k n - 2592 k n - 3780 k n - 7560 k n - 1848 k n 5 4 3 3 2 2 3 - 240 k - 142506 k n - 27084 k n - 95736 k n + 232116 k n 2 2 2 3 3 5 3 + 123114 k n + 32208 k n - 147712 k n - 2920 k - 6048 k n 8 3 5 4 6 7 + 2142 k n + 37772 n + 35329 n + 43855 n + 34874 n + 36624 n 5 2 2 2 4 6 - 48300 k n + 18004 n - 18768 n k + 3240 k + 224952 k n + 18872 k n 3 4 3 3 2 5 9 8 - 143304 k n - 164648 k n + 100212 k n + 5103 n + 21987 n 4 4 4 3 3 5 7 + 1320 k + 11244 k n + 52884 k n - 57708 k n + 16128 k n 2 6 4 4 4 2 / + 9450 k n + 36288 k n + 35712 k n ) / ((6 n + 1) (3 n + 1) / 6 (2 n + 1) (3 n + 2) (6 n + 5) (n + 1 - k) )]] 7 If the kernel is, binomial(n, k) , then the MWZ pair is the following consisting of a matrix of dimension, 7 7 [binomial(n, k) , [[1/2, 7/4 n + 3/2, 7/8 n (n + 1), 231 2 105 3 2 - --- n - 21/2 n - --- n - 5/2, - 7/32 n (n + 1) (47 n + 61 n + 20), 16 16 1463 5 777 2 5229 3 4 3/2 + ---- n + 14 n + --- n + ---- n + 273/4 n , 64 16 64 4 3 2 1 7/128 n (n + 1) (1361 n + 2722 n + 2019 n + 658 n + 80)], [- 7/2 -----, n + 1 3 2 40 + 47 n 40 + 107 n + 235 n + 170 n - 1/4 ---------, - 35/8 n + 5/4, 7/16 ----------------------------, n + 1 n + 1 2093 3 2625 2 385 ---- n + ---- n + --- n - 5/4, 32 32 16 2 4 3 5 96 + 3250 n + 920 n + 4650 n + 5535 n + 1557 n - 7/64 --------------------------------------------------, n + 1 63763 5 31871 4 94157 3 2 875 1 - ----- n - ----- n - ----- n - 945/4 n - --- n + 1/4], [21/2 --------, 128 32 128 32 2 (n + 1) 3 2 19 n + 16 53 n - 66 3566 n + 2275 n + 4975 n + 824 7/4 ---------, 1/8 ---------, - 1/16 --------------------------------, 2 n + 1 2 (n + 1) (n + 1) 3 2 773 n + 909 n + 206 n - 40 - 7/32 ----------------------------, n + 1 5 2 4 3 1984 + 19704 n + 34909 n + 71214 n + 104174 n + 123095 n 1/64 ------------------------------------------------------------, 2 (n + 1) 5 4 3 2 25467 n + 50760 n + 37149 n + 11636 n + 1236 n - 32 7/128 -------------------------------------------------------], [ n + 1 1 29 n + 24 3 n + 26 - 35/2 --------, - 7/4 ---------, 7/8 --------, 3 3 2 (n + 1) (n + 1) (n + 1) 2 3 1296 + 8193 n + 5778 n + 3781 n 1/16 ---------------------------------, 3 (n + 1) 3 2 6965 n + 7045 n + 398 n - 816 1/32 -------------------------------, 2 (n + 1) 5 3 2 4 8861 n + 448 + 30815 n + 17478 n + 4688 n + 26358 n - 7/64 -------------------------------------------------------, 3 (n + 1) 5 4 3 2 263627 n + 520576 n + 373429 n + 111372 n + 9804 n - 656 - 1/128 ------------------------------------------------------------], [ 2 (n + 1) 1 5 n + 4 23 n + 38 35/2 --------, 35/4 --------, - 7/8 ---------, 4 4 3 (n + 1) (n + 1) (n + 1) 2 3 160 + 1103 n + 754 n + 519 n - 7/16 ------------------------------, 4 (n + 1) 3 2 3953 n + 1933 n - 2358 n - 1248 - 1/32 ---------------------------------, 3 (n + 1) 5 2 3 4 13027 n + 6208 n + 24278 n + 44113 n + 38414 n + 544 5/64 --------------------------------------------------------, 4 (n + 1) 4 3 2 (3 n + 1) (9971 n + 15859 n + 7796 n + 844 n - 144) 7/128 ------------------------------------------------------], [ 3 (n + 1) 1 11 n + 8 29 n + 30 - 21/2 --------, - 7/4 --------, 7/8 ---------, 5 5 4 (n + 1) (n + 1) (n + 1) 2 3 3 2 64 + 545 n + 346 n + 269 n 51 n + 459 n + 506 n + 144 7/16 ----------------------------, - 7/32 ----------------------------, 5 4 (n + 1) (n + 1) 5 2 3 4 14960 n + 38629 n + 64094 n + 123223 n + 111294 n + 1088 - 1/64 ------------------------------------------------------------, 5 (n + 1) 5 4 3 2 69797 n + 120984 n + 65707 n + 6108 n - 4188 n - 816 - 1/128 --------------------------------------------------------], [ 4 (n + 1) 2 1 n 3 n + 2 n (75 n + 30 + 47 n ) 7/2 --------, 7/4 --------, - 35/8 --------, - 7/16 ---------------------, 6 6 5 6 (n + 1) (n + 1) (n + 1) (n + 1) 3 2 209 n + 409 n + 254 n + 48 7/32 ----------------------------, 5 (n + 1) 3 2 4 n (3590 n + 3515 n + 1510 n + 240 + 1361 n ) 7/64 ----------------------------------------------, 6 (n + 1) 5 4 3 2 10569 n + 29304 n + 31175 n + 15628 n + 3572 n + 272 - 1/128 --------------------------------------------------------]], 5 (n + 1) 7 2 3 4 5 6 k [1, k, k , k , k , k , k ], [- 1/2 ------------, 7 (n + 1 - k) 7 k (-224 n + 64 k - 256) - 1/128 ------------------------, 7 (n + 1 - k) 7 2 2 k (-112 n + 112 n + 64 k - 288 k - 224 n k + 224) 7 - 1/128 ----------------------------------------------------, - 1/128 k ( 7 (n + 1 - k) 3 2 2 3 2 840 n + 1232 n + 1960 n - 320 k + 448 k + 64 k + 336 n k - 112 n k 2 / 7 7 3 2 - 224 k n + 128) / (n + 1 - k) , - 1/128 k (2184 n + 700 n - 168 n / 4 4 3 2 2 + 1316 n + 64 k - 352 k + 672 k - 384 k + 1680 n k + 392 n k 3 2 2 2 3 / 7 7 + 840 k n + 560 k n - 112 k n - 224 k n) / (n + 1 - k) , - 1/128 k / 2 3 4 5 5 (-256 - 6692 n - 12250 n - 10052 n - 2926 n - 1792 n + 448 k + 64 k 2 3 4 3 3 2 4 3 - 960 k + 896 k - 384 k + 784 k n - 112 k n - 224 k n + 952 k n 2 4 2 3 2 2 2 / - 812 n k + 1316 k n + 840 k n - 560 k n + 1400 k n ) / / 7 7 2 5 (n + 1 - k) , - 1/128 k (112 - 28 n - 1722 n k - 528 k - 224 k n 5 4 3 3 2 2 3 2 2 - 416 k - 8526 k n - 1624 k n + 1120 k n - 280 k n - 1596 k n 2 3 3 3 5 4 + 1232 k n - 6734 k n - 1584 k - 12565 n - 25655 n - 25977 n 6 6 5 2 2 2 4 - 9527 n + 64 k - 2926 k n - 2856 n - 924 n k + 1232 k + 1316 k n 3 3 4 4 4 2 / 7 + 840 k n + 1120 k + 1008 k n - 112 k n ) / (n + 1 - k) ]] / 8 If the kernel is, binomial(n, k) , then the MWZ pair is the following consisting of a matrix of dimension, 7 2 8 n + 1 (n + 1) (12 n + 18 n + 7) [binomial(n, k) , [[7/2 -------, 7/4 --------------------------, 8 n + 7 (8 n + 7) (4 n + 3) 3 2 (n + 1) (48 n + 108 n + 81 n + 20) n 7/4 --------------------------------------, (8 n + 5) (8 n + 7) (4 n + 3) 4 3 2 2 (144 n + 408 n + 432 n + 201 n + 35) (n + 1) - 7/8 ------------------------------------------------, - 7/4 (8 n + 5) (8 n + 7) (4 n + 3) 6 5 4 3 2 (n + 1) (1056 n + 4296 n + 7236 n + 6450 n + 3204 n + 839 n + 90) n ------------------------------------------------------------------------, - (8 n + 5) (8 n + 7) (4 n + 3) (8 n + 3) 8 7 6 5 4 3 1/4 (960 n - 10944 n - 49428 n - 82191 n - 71739 n - 36000 n 2 2 - 10420 n - 1618 n - 105) (n + 1) /((8 n + 3) (4 n + 3) (8 n + 7) 9 8 (4 n + 1) (8 n + 5)), 1/8 n (3 n + 2) (n + 1) (259584 n + 1223808 n 7 6 5 4 3 2 + 2528448 n + 2994936 n + 2231880 n + 1079421 n + 336644 n + 64767 n + 6886 n + 300)/((8 n + 3) (4 n + 3) (8 n + 1) (8 n + 7) (4 n + 1) 2 27 308 n + 456 n + 175 (8 n + 5))], [- -------, - 1/2 --------------------, 8 n + 7 (8 n + 7) (4 n + 3) 4 3 2 336 n + 672 n + 380 n - 35 - 3/2 -----------------------------, (8 n + 5) (8 n + 7) (4 n + 3) 4 3 2 (13 n + 7) (n + 1) (336 n + 924 n + 945 n + 422 n + 70) 1/2 ----------------------------------------------------------, 1/2 ( (8 n + 5) (8 n + 7) (2 n + 1) (4 n + 3) 8 7 6 5 4 3 55104 n + 249312 n + 478548 n + 503580 n + 312020 n + 112388 n 2 + 20902 n + 1162 n - 105)/((8 n + 5) (8 n + 7) (2 n + 1) (4 n + 3) 9 8 7 6 (8 n + 3)), - 1/2 (n + 1) (21888 n + 217440 n + 700776 n + 1144116 n 5 4 3 2 + 1099530 n + 659622 n + 249761 n + 57945 n + 7522 n + 420)/((8 n + 3) 3 (4 n + 3) (2 n + 1) (8 n + 7) (4 n + 1) (8 n + 5)), - 1/2 (327916 n 9 5 7 8 + 802 n - 15 + 76100736 n + 9447496 n + 55418628 n + 78310614 n 2 6 4 12 10 + 27088 n + 27425530 n + 2204926 n + 3122688 n + 48571104 n 11 + 18365952 n )/((8 n + 3) (4 n + 3) (8 n + 1) (2 n + 1) (8 n + 7) 2 90 64 n + 93 n + 35 (4 n + 1) (8 n + 5))], [-----------------, 15/2 ---------------------------, (n + 1) (8 n + 7) (8 n + 7) (4 n + 3) (n + 1) 4 3 2 448 n + 652 n - 34 n - 403 n - 147 5/2 -------------------------------------, (8 n + 5) (8 n + 7) (4 n + 3) (n + 1) 5 4 3 2 10752 n + 34776 n + 44572 n + 28174 n + 8786 n + 1085 - 3/4 ---------------------------------------------------------, - ( (8 n + 5) (8 n + 7) (2 n + 1) (4 n + 3) 8 7 6 5 4 3 86016 n + 382536 n + 716898 n + 728095 n + 425685 n + 136906 n 2 + 18276 n - 1127 n - 420)/((8 n + 3) (4 n + 3) (2 n + 1) (8 n + 7) 9 8 7 6 (8 n + 5) (n + 1)), 3/4 (129024 n + 856128 n + 2306976 n + 3395112 n 5 4 3 2 + 3043300 n + 1733324 n + 629008 n + 140476 n + 17572 n + 945)/( (8 n + 3) (4 n + 3) (2 n + 1) (8 n + 7) (4 n + 1) (8 n + 5)), 1/2 ( 12 11 8 2 6 10727424 n + 62985600 n + 264206748 n + 70499 n + 90201275 n 4 7 10 5 + 6883406 n + 184980348 n - 120 + 166017888 n + 890 n + 30432293 n 9 3 + 258737016 n + 969623 n )/((8 n + 3) (4 n + 3) (8 n + 1) (2 n + 1) 168 (8 n + 7) (4 n + 1) (8 n + 5) (n + 1))], [- ------------------, 2 (8 n + 7) (n + 1) 2 68 n + 96 n + 35 -12 ----------------------------, 2 (8 n + 7) (4 n + 3) (n + 1) 4 3 2 288 n - 228 n - 1449 n - 1318 n - 357 -3 ----------------------------------------, 2 (8 n + 5) (8 n + 7) (4 n + 3) (n + 1) 5 4 3 2 32704 n + 103496 n + 129402 n + 79474 n + 23939 n + 2835 1/2 ------------------------------------------------------------, ( (8 n + 5) (8 n + 7) (2 n + 1) (4 n + 3) (n + 1) 8 7 6 5 4 3 142464 n + 613872 n + 1099244 n + 1038816 n + 531124 n + 119528 n 2 / - 8889 n - 9722 n - 1365) / ((8 n + 5) (8 n + 7) (2 n + 1) (4 n + 3) / 2 9 8 7 6 (8 n + 3) (n + 1) ), - (317184 n + 1821792 n + 4455360 n + 6105314 n 5 4 3 2 + 5171185 n + 2806077 n + 973814 n + 207943 n + 24771 n + 1260)/( (8 n + 3) (4 n + 3) (2 n + 1) (8 n + 7) (4 n + 1) (8 n + 5) (n + 1)), - 1/2 12 11 8 9 (20256768 n + 118341888 n + 481493060 n + 477874992 n - 399 - 3196 n 2 6 10 7 5 + 71703 n + 157523906 n + 309685824 n + 331101072 n + 51296960 n 3 4 / + 1387394 n + 10970768 n ) / ((8 n + 5) (4 n + 1) (8 n + 7) (2 n + 1) / 2 189 (8 n + 1) (4 n + 3) (8 n + 3) (n + 1) )], [------------------, 3 (8 n + 7) (n + 1) 2 76 n + 102 n + 35 21/2 ----------------------------, 3 (8 n + 7) (4 n + 3) (n + 1) 4 3 2 144 n + 1236 n + 2243 n + 1536 n + 364 - 9/2 -----------------------------------------, 3 (8 n + 5) (8 n + 7) (4 n + 3) (n + 1) 5 4 3 2 25824 n + 79200 n + 95460 n + 56075 n + 15951 n + 1750 - 3/4 ----------------------------------------------------------, - 1/2 ( 2 (8 n + 5) (8 n + 7) (2 n + 1) (4 n + 3) (n + 1) 8 7 6 5 4 3 255808 n + 1031472 n + 1662136 n + 1284394 n + 360376 n - 143555 n 2 / - 143712 n - 42407 n - 4515) / ((8 n + 5) (8 n + 7) (2 n + 1) (4 n + 3) / 3 9 8 7 6 (8 n + 3) (n + 1) ), 3/4 (715008 n + 3787392 n + 8662768 n + 11202572 n 5 4 3 2 / + 9000470 n + 4641140 n + 1527940 n + 307535 n + 34070 n + 1575) / ( / 2 (8 n + 3) (4 n + 3) (2 n + 1) (8 n + 7) (4 n + 1) (8 n + 5) (n + 1) ), 1/4 10 3 7 4 (660734592 n - 21316 n + 1575962 n + 654049460 n + 17040619 n 9 12 11 8 2 + 1000351088 n + 44362752 n + 256236288 n + 983200808 n - 35278 n 6 5 / + 297089851 n + 90194848 n - 1344) / ((8 n + 5) (4 n + 1) (8 n + 7) / 3 126 (2 n + 1) (8 n + 1) (4 n + 3) (8 n + 3) (n + 1) )], [- ------------------, 4 (8 n + 7) (n + 1) 4 3 2 (10 n + 7) (2 n + 1) 80 n + 344 n + 490 n + 292 n + 63 -21 ----------------------------, 21 --------------------------------------, 4 4 (8 n + 7) (4 n + 3) (n + 1) (8 n + 5) (8 n + 7) (4 n + 3) (n + 1) 5 4 3 2 4272 n + 12444 n + 14085 n + 7625 n + 1929 n + 175 8 3 ------------------------------------------------------, 3 (17344 n 3 (8 n + 5) (8 n + 7) (2 n + 1) (4 n + 3) (n + 1) 7 6 5 4 3 2 + 56640 n + 55228 n - 17914 n - 80493 n - 71564 n - 31202 n - 6936 n / 4 - 630) / ((8 n + 5) (8 n + 7) (2 n + 1) (4 n + 3) (8 n + 3) (n + 1) ), - / 9 8 7 6 5 1/2 (969472 n + 4824384 n + 10397680 n + 12684232 n + 9601804 n 4 3 2 / + 4644754 n + 1421193 n + 260959 n + 25377 n + 945) / ((8 n + 3) / 3 (4 n + 3) (2 n + 1) (8 n + 7) (4 n + 1) (8 n + 5) (n + 1) ), - 3/2 (-189 7 12 6 4 5 + 51976036 n + 4408320 n + 21106510 n + 570879 n + 5235262 n 10 11 8 2 9 + 62030656 n + 24826368 n + 84099516 n - 31022 n + 90216768 n 3 / - 4002 n - 66392 n ) / ((8 n + 5) (4 n + 1) (8 n + 7) (2 n + 1) / 4 42 (8 n + 1) (4 n + 3) (8 n + 3) (n + 1) )], [------------------, 5 (8 n + 7) (n + 1) 3 2 n 12 n + 33 n + 27 n + 7 21 ------------------, -21 ----------------------------, 5 5 (8 n + 7) (n + 1) (8 n + 7) (8 n + 5) (n + 1) 2 (44 n + 60 n + 21) n - 21/2 ----------------------------, 4 (8 n + 7) (8 n + 5) (n + 1) 6 5 4 3 2 40 n - 1082 n - 3468 n - 4142 n - 2373 n - 654 n - 70 -3 ----------------------------------------------------------, 5 (8 n + 5) (8 n + 7) (8 n + 3) (n + 1) 5 4 3 2 (4056 n + 12750 n + 15825 n + 9680 n + 2920 n + 350) n 3/2 ----------------------------------------------------------, 1/2 ( 4 (8 n + 5) (8 n + 7) (8 n + 3) (n + 1) 8 2 6 4 3 9 369504 n - 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5587968 k n - 17896200 k n - 41916168 k n - 19808592 k n 2 5 9 8 10 11 + 50915472 k n + 17947248 n + 25720116 n + 7257600 n + 1215744 n 4 4 4 3 3 5 7 + 25830 k + 469848 k n + 11114152 k n - 52588848 k n - 7635984 k n 2 6 4 4 4 2 / + 33553152 k n + 21983536 k n + 3197252 k n ) / ((8 n + 1) / 8 (4 n + 1) (8 n + 3) (2 n + 1) (8 n + 5) (4 n + 3) (8 n + 7) (n + 1 - k) ), 8 4 5 2 1/8 k (-4176 - 71160 n - 77418112 k n + 3080608 n k + 27108 k 5 2 5 4 2 7 3 6 + 5082144 k n + 28915392 k n - 15675216 k n + 71265744 k n 5 5 4 3 3 2 + 870680 k n + 60900 k + 24898952 k n + 1834744 k n + 11768504 k n 2 3 2 2 2 3 - 29002280 k n - 8113788 k n - 1224008 k n + 11359472 k n 3 5 3 8 3 5 + 117516 k + 15861056 k n + 18137640 k n - 1783488 n - 1804443 n 4 4 6 9 3 7 - 3426579 n - 53411328 k n + 16059264 k n + 20703168 k n 6 7 2 8 6 7 + 8698683 n + 29919609 n - 241248 k n - 18900 k + 2520 k 10 5 6 2 9 3 8 + 8347776 k n + 19447296 k n - 2937984 k n - 3822336 k n 12 5 2 10 4 8 + 10748160 n + 33646384 k n - 3515904 k n - 1892352 k n 2 2 11 6 3 - 492096 n + 451272 n k - 75768 k + 1708032 k n - 4044576 k n 6 2 6 4 3 9 5 7 - 1402512 k n - 6637440 k n - 3677184 k n + 6119424 k n 6 5 6 6 6 2 4 - 6190080 k n - 3022848 k n - 255960 k n - 61090980 k n 4 7 5 5 3 10 5 8 - 18809088 k n + 31341440 k n - 516096 k n + 688128 k n 4 9 7 2 7 4 7 3 + 344064 k n + 163328 k n + 604160 k n + 426240 k n 2 11 12 7 5 7 6 7 - 946176 k n - 30720 k n + 442368 k n + 131072 k n + 32112 k n 6 7 13 6 3 4 - 589824 k n + 1557504 n + 29224176 k n + 83510368 k n 3 3 2 5 9 8 + 40767728 k n - 76641296 k n + 67740936 n + 54632454 n 10 11 4 4 4 3 + 58057104 n + 32654592 n - 109200 k - 1637080 k n - 33388744 k n 3 5 7 2 6 4 4 + 102695664 k n + 20461056 k n - 53389272 k n - 65426364 k n 4 2 / - 10069716 k n ) / ((8 n + 1) (4 n + 1) (8 n + 3) (2 n + 1) (8 n + 5) / 8 (4 n + 3) (8 n + 7) (n + 1 - k) )]]