---------------------------------------------- Using Doron Zeilberger's Maple package AsyRec, Shalosh B. Ekhad found out th\ at the asympotics to order, 10 of the sequence enumerating positive rook walks from, [0, 0], to , [n, n], in dimension , 2, is as follows: 1/2 1/2 n 1/2 / 5 11 275 78141 4 2 9 (1/n) |1 - ---- - ------- - -------- - ---------- | 32 n 2 3 4 \ 2048 n 65536 n 8388608 n 3643395 389989039 29103377875 21306877320157 - ------------ - -------------- - --------------- - ------------------ 5 6 7 8 268435456 n 17179869184 n 549755813888 n 140737488355328 n 2212481004634615 520387639958664933 \ / 1/2 - ------------------- - ----------------------| / (6 Pi ) 9 10| / 4503599627370496 n 288230376151711744 n / where the constant in front was deduced by M. Erickson, Suren Fernando, and \ Khang Tran from the work of Robin Pemantle and Mark C. Wilson, who can easily do the le\ ading asymptotics but probably would have to work much harder to get order, 10 ---------------------------------------------- Using Doron Zeilberger's Maple package AsyRec, Shalosh B. Ekhad found out th\ at the asympotics to order, 10 of the sequence enumerating positive rook walks from, [0, 0, 0], to , [n, n, n], in dimension , 3, is as follows: 1/2 n / 98 1582 230722 10524818 9 3 64 |1 - ----- + -------- + ----------- - ------------- | 375 n 2 3 4 \ 84375 n 94921875 n 2373046875 n 59410460078 33517945968458 3119023030652354 - ----------------- - -------------------- - ---------------------- 5 6 7 13348388671875 n 15016937255859375 n 1126270294189453125 n 11029396169407144774 372193214725249512886 - ------------------------- - -------------------------- 8 9 1267054080963134765625 n 19005811214447021484375 n 1399208649796334323345166 \ - ------------------------------|/(40 Pi n) 10| 35635896027088165283203125 n / where the constant in front was deduced by M. Erickson, Suren Fernando, and \ Khang Tran from the work of Robin Pemantle and Mark C. Wilson, who can easily do the le\ ading asymptotics but probably would have to work much harder to get order, 10 ---------------------------------------------- Using Doron Zeilberger's Maple package AsyRec, Shalosh B. Ekhad found out th\ at the asympotics to order, 10 of the sequence enumerating positive rook walks from, [0, 0, 0, 0], to , [n, n, n, n], in dimension , 4, is as follows: 1/2 1/2 n 3/2 / 305 1025 96643 51398095 4 6 2 625 (1/n) |1 - ----- + -------- + ----------- - -------------- | 864 n 2 3 4 \ 20736 n 26873856 n 11609505792 n 415699781 131960799137 372530534021633 - --------------- + ------------------ + --------------------- 5 6 7 123834728448 n 320979616137216 n 277326388342554624 n 55805555347657349 1089602772737060868377 - ----------------------- - --------------------------- 8 9 26623333280885243904 n 207023039592163656597504 n 436512405078536092202687 \ / 3/2 - -------------------------------| / (45 Pi ) 10| / 178867906207629399300243456 n / where the constant in front was deduced by M. Erickson, Suren Fernando, and \ Khang Tran from the work of Robin Pemantle and Mark C. Wilson, who can easily do the le\ ading asymptotics but probably would have to work much harder to get order, 10 ---------------------------------------------- Using Doron Zeilberger's Maple package AsyRec, Shalosh B. Ekhad found out th\ at the asympotics to order, 10 of the sequence enumerating positive rook walks from, [0, 0, 0, 0, 0], to , [n, n, n, n, n], in dimension , 5, is as follows: 1/2 n / 754 1262602 17891798 1161815848486 125 5 7776 |1 - ------ + ----------- + -------------- - ------------------ | 1715 n 2 3 4 \ 14706125 n 18015003125 n 216270112515625 n 4202055473912266 3252537039912472358 - ---------------------- + ------------------------- 5 6 1324654439158203125 n 2271782363156318359375 n 46067928295450510589806 267103734384353011891873642 + ----------------------------- - --------------------------------- 7 8 19480533764065429931640625 n 167045577026861061663818359375 n 49712987676772694941922786122 - ----------------------------------- 9 11459326584042668830137939453125 n 434720893519172981430438325878466 \ / 2 2 + ----------------------------------------| / (1176 Pi n ) 10| / 350941876636306732922974395751953125 n / where the constant in front was deduced by M. Erickson, Suren Fernando, and \ Khang Tran from the work of Robin Pemantle and Mark C. Wilson, who can easily do the le\ ading asymptotics but probably would have to work much harder to get order, 10 ---------------------------------------------- Using Doron Zeilberger's Maple package AsyRec, Shalosh B. Ekhad found out th\ at the asympotics to order, 10 of the sequence enumerating positive rook walks from, [0, 0, 0, 0, 0, 0], to , [n, n, n, n, n, n], in dimension , 6, is as follows: 1/2 1/2 n 5/2 / 805 113435 2097179 81 8 2 117649 (1/n) |1 - ------ + --------- - ------------ | 1536 n 2 3 \ 884736 n 402653184 n 218919309 480052172995 1597596305193161 - -------------- - ------------------ + --------------------- 4 5 6 34359738368 n 158329674399744 n 729583139634020352 n 6368167288440313189 9927946266749471925923 + ------------------------- - ---------------------------- 7 8 2241279404955710521344 n 5163907749017957041176576 n 111141252363884997305895515 7575880526064060720598308793 \ - -------------------------------- + -----------------------------------| 9 10| 23795286907474746045741662208 n 3045796724156767493854932762624 n / / 5/2 / (1792 Pi ) / where the constant in front was deduced by M. Erickson, Suren Fernando, and \ Khang Tran from the work of Robin Pemantle and Mark C. Wilson, who can easily do the le\ ading asymptotics but probably would have to work much harder to get order, 10 ---------------------------------------------- Using Doron Zeilberger's Maple package AsyRec, Shalosh B. Ekhad found out th\ at the asympotics to order, 10 of the sequence enumerating positive rook walks from, [0, 0, 0, 0, 0, 0, 0], to , [n, n, n, n, n, n, n], in dimension , 7, is as follows: 1/2 n / 3100 32215864 33175681844 2401 7 2097152 |1 - ------ + ------------ - ---------------- | 5103 n 2 3 \ 182284263 n 2170458719541 n 541406380075600 1098029757346162436 - -------------------- - ------------------------ 4 5 77530955920724061 n 395640468063454883283 n 13634511683213909596408 3679142151039207278248200884 + ---------------------------- + ---------------------------------- 6 7 4710891053231557295250681 n 1177944175187391207005547032007 n 14766906338353457226854086837952 - ------------------------------------- 8 6011049125981257329349306504331721 n 122989510171050312127949019146950876 - ----------------------------------------- 9 23857853981019610340187397515692600649 n 2961840244283043794913518990253476607112 \ / 3 3 + ----------------------------------------------| / (46656 Pi n ) 10| / 852226402056001500961834026658055387782929 n / where the constant in front was deduced by M. Erickson, Suren Fernando, and \ Khang Tran from the work of Robin Pemantle and Mark C. Wilson, who can easily do the le\ ading asymptotics but probably would have to work much harder to get order, 10 ---------------------------------------------- Using Doron Zeilberger's Maple package AsyRec, Shalosh B. Ekhad found out th\ at the asympotics to order, 10 of the sequence enumerating positive rook walks from, [0, 0, 0, 0, 0, 0, 0, 0], to , [n, n, n, n, n, n, n, n], in dimension , 8, is as follows: 1/2 1/2 n 7/2 / 5523 11857923 1518902063 128 10 2 43046721 (1/n) |1 - ------ + ----------- - -------------- | 8000 n 2 3 \ 51200000 n 51200000000 n 699681887457 244692952164069 11445471958779097 - ------------------ - --------------------- + ---------------------- 4 5 6 102400000000000 n 102400000000000000 n 3200000000000000000 n 860062935073809074247 12920554723870911295244451 + --------------------------- - ------------------------------- 7 8 262144000000000000000000 n 4194304000000000000000000000 n 1881907820343363765758684887603 - ------------------------------------ 9 335544320000000000000000000000000 n 24106525391591538325400900852748099 \ / 7/2 + -----------------------------------------| / (5625 Pi ) 10| / 5368709120000000000000000000000000000 n / where the constant in front was deduced by M. Erickson, Suren Fernando, and \ Khang Tran from the work of Robin Pemantle and Mark C. Wilson, who can easily do the le\ ading asymptotics but probably would have to work much harder to get order, 10 ---------------------------------------------- Using Doron Zeilberger's Maple package AsyRec, Shalosh B. Ekhad found out th\ at the asympotics to order, 10 of the sequence enumerating positive rook walks from, [0, 0, 0, 0, 0, 0, 0, 0, 0], to , [n, n, n, n, n, n, n, n, n], in dimension , 9, is as follows: 1/2 n / 9260 378331400 6800933928716 59049 9 1000000000 |1 - ------- + ------------- - ------------------ | 11979 n 2 3 \ 1291467969 n 139234453205859 n 2516929280715920 94188330000143955652 - --------------------- - -------------------------- 4 5 454878958623541353 n 49040955408162616808283 n 82038818728564611725831288 2320890643543191282303949136108 + -------------------------------- + ------------------------------------ 6 7 19386232959534539562631927881 n 696683053866792748262303592259497 n 2332059174423099111286305953686208 - --------------------------------------- 8 620744600995312338701712500703211827 n 145428067097494334296408495677356024888084 - ----------------------------------------------- 9 24293083912579739532840628488032971211893011 n 541225062698518916007137563273458796928092376 \ / 4 4 + ---------------------------------------------------| / (2342560 Pi n 10| / 97002284062930899954632629552715654049088792923 n / ) where the constant in front was deduced by M. Erickson, Suren Fernando, and \ Khang Tran from the work of Robin Pemantle and Mark C. Wilson, who can easily do the le\ ading asymptotics but probably would have to work much harder to get order, 10 --------------------------------------------------------------- This took, 255.574, seconds of CPU time (using the already computed Manuel K\ auers' computer recurrences