Explicit Expressions for the coefficient of x^n in, /infinity \r | ----- / n \| | \ | 2 (4 n + 1)! x || 2 (4 n + 1)! | ) |-------------------|| , divided by, -------------------, | / \(n + 1)! (3 n + 2)!/| (n + 1)! (3 n + 2)! | ----- | \ n = 1 / for r from 2 to, 18 By Shalosh B. Ekhad Theorem: /infinity \r | ----- / n \| | \ | 2 (4 n + 1)! x || Let A[r](n) be the coefficient of x^n in , | ) |-------------------|| , | / \(n + 1)! (3 n + 2)!/| | ----- | \ n = 1 / 2 (4 n + 1)! divided by , ------------------- (n + 1)! (3 n + 2)! Let B[r] be the limit of A[r][n] as n goes to infinity then we have the following A[2, n], = 2 10 (n - 1) (n + 14 n + 12) ----------------------------- 3 (3 n + 5) (3 n + 4) (n + 2) B[2], = 10 -- 27 and in Maple notation 10/3*(n-1)*(n^2+14*n+12)/(3*n+5)/(3*n+4)/(n+2) 10/27 A[3, n], = 4 3 2 5 (n - 1) (n - 2) (5 n + 160 n + 1803 n + 3768 n + 2016) ----------------------------------------------------------- 3 (3 n + 8) (3 n + 5) (3 n + 7) (3 n + 4) (n + 3) (n + 2) B[3] = 25 --- 243 and in Maple notation 5/3*(n-1)*(n-2)*(5*n^4+160*n^3+1803*n^2+3768*n+2016)/(3*n+8)/(3*n+5)/(3*n+7)/(3 *n+4)/(n+3)/(n+2) 25/243 A[4, n], = 20 (n - 1) (n - 2) (n - 3) 6 5 4 3 2 (25 n + 1350 n + 31495 n + 347406 n + 1211092 n + 1580304 n + 665280)/ (27 (3 n + 11) (3 n + 8) (3 n + 5) (3 n + 10) (3 n + 7) (3 n + 4) (n + 4) (n + 3) (n + 2)) B[4], = 500 ----- 19683 and in Maple notation 20/27*(n-1)*(n-2)*(n-3)*(25*n^6+1350*n^5+31495*n^4+347406*n^3+1211092*n^2+ 1580304*n+665280)/(3*n+11)/(3*n+8)/(3*n+5)/(3*n+10)/(3*n+7)/(3*n+4)/(n+4)/(n+3) /(n+2) 500/19683 A[5, n], = 8 7 6 5 25 (n - 1) (n - 2) (n - 3) (n - 4) (125 n + 10000 n + 371450 n + 7831720 n 4 3 2 + 90056453 n + 442852216 n + 996421764 n + 1007115984 n + 363242880)/( 81 (3 n + 14) (3 n + 11) (3 n + 8) (3 n + 5) (3 n + 13) (3 n + 10) (3 n + 7) (3 n + 4) (n + 5) (n + 4) (n + 3) (n + 2)) B[5], = 3125 ------ 531441 and in Maple notation 25/81*(n-1)*(n-2)*(n-3)*(n-4)*(125*n^8+10000*n^7+371450*n^6+7831720*n^5+ 90056453*n^4+442852216*n^3+996421764*n^2+1007115984*n+363242880)/(3*n+14)/(3*n+ 11)/(3*n+8)/(3*n+5)/(3*n+13)/(3*n+10)/(3*n+7)/(3*n+4)/(n+5)/(n+4)/(n+3)/(n+2) 3125/531441 A[6, n], = 10 9 8 10 (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (625 n + 68750 n + 3628750 n 7 6 5 4 + 116901500 n + 2395396825 n + 29421692462 n + 186231365400 n 3 2 + 617582906760 n + 1076423055120 n + 918818062848 n + 296406190080)/(81 (3 n + 17) (3 n + 14) (3 n + 11) (3 n + 8) (3 n + 5) (3 n + 16) (3 n + 13) (3 n + 10) (3 n + 7) (3 n + 4) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2)) B[6], = 6250 ------- 4782969 and in Maple notation 10/81*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(625*n^10+68750*n^9+3628750*n^8+116901500*n ^7+2395396825*n^6+29421692462*n^5+186231365400*n^4+617582906760*n^3+ 1076423055120*n^2+918818062848*n+296406190080)/(3*n+17)/(3*n+14)/(3*n+11)/(3*n+ 8)/(3*n+5)/(3*n+16)/(3*n+13)/(3*n+10)/(3*n+7)/(3*n+4)/(n+6)/(n+5)/(n+4)/(n+3)/( n+2) 6250/4782969 A[7, n], = 12 11 175 (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (625 n + 90000 n 10 9 8 7 + 6325625 n + 281268000 n + 8516307975 n + 175323137472 n 6 5 4 + 2320161434435 n + 17886472474464 n + 79792191915340 n 3 2 + 206727295234608 n + 302645014842240 n + 228319317305856 n + 67580611338240)/(729 (3 n + 20) (3 n + 17) (3 n + 14) (3 n + 11) (3 n + 8) (3 n + 5) (3 n + 19) (3 n + 16) (3 n + 13) (3 n + 10) (3 n + 7) (3 n + 4) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2)) B[7], = 109375 --------- 387420489 and in Maple notation 175/729*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*(625*n^12+90000*n^11+6325625*n^10+ 281268000*n^9+8516307975*n^8+175323137472*n^7+2320161434435*n^6+17886472474464* n^5+79792191915340*n^4+206727295234608*n^3+302645014842240*n^2+228319317305856* n+67580611338240)/(3*n+20)/(3*n+17)/(3*n+14)/(3*n+11)/(3*n+8)/(3*n+5)/(3*n+19)/ (3*n+16)/(3*n+13)/(3*n+10)/(3*n+7)/(3*n+4)/(n+7)/(n+6)/(n+5)/(n+4)/(n+3)/(n+2) 109375/387420489 A[8, n], = 14 200 (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (n - 7) (3125 n 13 12 11 10 + 568750 n + 51008125 n + 2952031250 n + 120616941375 n 9 8 7 + 3564549572010 n + 75011757331015 n + 1071261463254878 n 6 5 4 + 9703709965152728 n + 54870097208217592 n + 193681878688205040 n 3 2 + 421496433468194112 n + 541984884627439872 n + 371487057175922688 n + 102587368011448320)/(2187 (3 n + 23) (3 n + 20) (3 n + 17) (3 n + 14) (3 n + 11) (3 n + 8) (3 n + 5) (3 n + 22) (3 n + 19) (3 n + 16) (3 n + 13) (3 n + 10) (3 n + 7) (3 n + 4) (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2)) B[8], = 625000 ----------- 10460353203 and in Maple notation 200/2187*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*(n-7)*(3125*n^14+568750*n^13+ 51008125*n^12+2952031250*n^11+120616941375*n^10+3564549572010*n^9+ 75011757331015*n^8+1071261463254878*n^7+9703709965152728*n^6+54870097208217592* n^5+193681878688205040*n^4+421496433468194112*n^3+541984884627439872*n^2+ 371487057175922688*n+102587368011448320)/(3*n+23)/(3*n+20)/(3*n+17)/(3*n+14)/(3 *n+11)/(3*n+8)/(3*n+5)/(3*n+22)/(3*n+19)/(3*n+16)/(3*n+13)/(3*n+10)/(3*n+7)/(3* n+4)/(n+8)/(n+7)/(n+6)/(n+5)/(n+4)/(n+3)/(n+2) 625000/10460353203 A[9, n], = 16 25 (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (n - 7) (n - 8) (15625 n 15 14 13 12 + 3500000 n + 388512500 n + 28157710000 n + 1471455483750 n 11 10 9 + 57589291846800 n + 1695686514421100 n + 36823499062435120 n 8 7 6 + 566623013737989961 n + 5883512920147660112 n + 40461585694240387992 n 5 4 + 183663017691739672800 n + 546422776901073190224 n 3 2 + 1041865968637322656128 n + 1209989486787870567168 n + 767428785578256721920 n + 200045367622324224000)/(729 (3 n + 26) (3 n + 23) (3 n + 20) (3 n + 17) (3 n + 14) (3 n + 11) (3 n + 8) (3 n + 5) (3 n + 25) (3 n + 22) (3 n + 19) (3 n + 16) (3 n + 13) (3 n + 10) (3 n + 7) (3 n + 4) (n + 9) (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2)) B[9], = 390625 ----------- 31381059609 and in Maple notation 25/729*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*(n-7)*(n-8)*(15625*n^16+3500000*n^15 +388512500*n^14+28157710000*n^13+1471455483750*n^12+57589291846800*n^11+ 1695686514421100*n^10+36823499062435120*n^9+566623013737989961*n^8+ 5883512920147660112*n^7+40461585694240387992*n^6+183663017691739672800*n^5+ 546422776901073190224*n^4+1041865968637322656128*n^3+1209989486787870567168*n^2 +767428785578256721920*n+200045367622324224000)/(3*n+26)/(3*n+23)/(3*n+20)/(3*n +17)/(3*n+14)/(3*n+11)/(3*n+8)/(3*n+5)/(3*n+25)/(3*n+22)/(3*n+19)/(3*n+16)/(3*n +13)/(3*n+10)/(3*n+7)/(3*n+4)/(n+9)/(n+8)/(n+7)/(n+6)/(n+5)/(n+4)/(n+3)/(n+2) 390625/31381059609 A[10, n], = 250 (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (n - 7) (n - 8) (n - 9) ( 18 17 16 15 78125 n + 21093750 n + 2832000000 n + 250057575000 n 14 13 12 + 16129151298750 n + 795938987686500 n + 30534485772468500 n 11 10 + 906798973422904200 n + 20427547655336548845 n 9 8 + 337686950884283521398 n + 3947955480810255108516 n 7 6 + 32043500652066554061936 n + 179512767333384682604720 n 5 4 + 690539558794947366099552 n + 1800054096927431745057984 n 3 2 + 3089887924189906537726464 n + 3302805644601397253544960 n + 1963602485859499681996800 n + 487310515527981809664000)/(19683 (3 n + 29) (3 n + 26) (3 n + 23) (3 n + 20) (3 n + 17) (3 n + 14) (3 n + 11) (3 n + 8) (3 n + 5) (3 n + 28) (3 n + 25) (3 n + 22) (3 n + 19) (3 n + 16) (3 n + 13) (3 n + 10) (3 n + 7) (3 n + 4) (n + 10) (n + 9) (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2)) B[10], = 19531250 ------------- 7625597484987 and in Maple notation 250/19683*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*(n-7)*(n-8)*(n-9)*(78125*n^18+ 21093750*n^17+2832000000*n^16+250057575000*n^15+16129151298750*n^14+ 795938987686500*n^13+30534485772468500*n^12+906798973422904200*n^11+ 20427547655336548845*n^10+337686950884283521398*n^9+3947955480810255108516*n^8+ 32043500652066554061936*n^7+179512767333384682604720*n^6+ 690539558794947366099552*n^5+1800054096927431745057984*n^4+ 3089887924189906537726464*n^3+3302805644601397253544960*n^2+ 1963602485859499681996800*n+487310515527981809664000)/(3*n+29)/(3*n+26)/(3*n+23 )/(3*n+20)/(3*n+17)/(3*n+14)/(3*n+11)/(3*n+8)/(3*n+5)/(3*n+28)/(3*n+25)/(3*n+22 )/(3*n+19)/(3*n+16)/(3*n+13)/(3*n+10)/(3*n+7)/(3*n+4)/(n+10)/(n+9)/(n+8)/(n+7)/ (n+6)/(n+5)/(n+4)/(n+3)/(n+2) 19531250/7625597484987 A[11, n], = 275 (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (n - 7) (n - 8) (n - 9) 20 19 18 17 (n - 10) (390625 n + 125000000 n + 19932421875 n + 2100058125000 n 16 15 14 + 162998007281250 n + 9812713709640000 n + 468929848263193750 n 13 12 + 17882056172995670000 n + 539717687867022031125 n 11 10 + 12647249945053650458400 n + 223840869361781977857207 n 9 8 + 2906822815545027467301000 n + 27223159945086515727267640 n 7 6 + 182549042700526995526230560 n + 872351065103429658321105456 n 5 4 + 2945228241209805686343502080 n + 6902569938133543227991752960 n 3 2 + 10868899678282135763981015040 n + 10838203759365349592131264512 n + 6098034294882387606412984320 n + 1450236094211273865560064000)/(59049 (3 n + 32) (3 n + 29) (3 n + 26) (3 n + 23) (3 n + 20) (3 n + 17) (3 n + 14) (3 n + 11) (3 n + 8) (3 n + 5) (3 n + 31) (3 n + 28) (3 n + 25) (3 n + 22) (3 n + 19) (3 n + 16) (3 n + 13) (3 n + 10) (3 n + 7) (3 n + 4) (n + 11) (n + 10) (n + 9) (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2)) B[11], = 107421875 --------------- 205891132094649 and in Maple notation 275/59049*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*(n-7)*(n-8)*(n-9)*(n-10)*(390625* n^20+125000000*n^19+19932421875*n^18+2100058125000*n^17+162998007281250*n^16+ 9812713709640000*n^15+468929848263193750*n^14+17882056172995670000*n^13+ 539717687867022031125*n^12+12647249945053650458400*n^11+ 223840869361781977857207*n^10+2906822815545027467301000*n^9+ 27223159945086515727267640*n^8+182549042700526995526230560*n^7+ 872351065103429658321105456*n^6+2945228241209805686343502080*n^5+ 6902569938133543227991752960*n^4+10868899678282135763981015040*n^3+ 10838203759365349592131264512*n^2+6098034294882387606412984320*n+ 1450236094211273865560064000)/(3*n+32)/(3*n+29)/(3*n+26)/(3*n+23)/(3*n+20)/(3*n +17)/(3*n+14)/(3*n+11)/(3*n+8)/(3*n+5)/(3*n+31)/(3*n+28)/(3*n+25)/(3*n+22)/(3*n +19)/(3*n+16)/(3*n+13)/(3*n+10)/(3*n+7)/(3*n+4)/(n+11)/(n+10)/(n+9)/(n+8)/(n+7) /(n+6)/(n+5)/(n+4)/(n+3)/(n+2) 107421875/205891132094649 A[12, n], = 500 (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (n - 7) (n - 8) (n - 9) 22 21 20 (n - 10) (n - 11) (390625 n + 146093750 n + 27263671875 n 19 18 17 + 3371893593750 n + 308936639343750 n + 22156447588027500 n 16 15 + 1279052684751943750 n + 60115193382267897500 n 14 13 + 2299404607657354894125 n + 70878598177548115189950 n 12 11 + 1729900820671159605447927 n + 32666103637559910020696622 n 10 9 + 466201654576472678847544924 n + 4950647702507197180824497504 n 8 7 + 38811779597851432703525662800 n + 223575427080831090549810361440 n 6 5 + 940368325304867050278049974336 n + 2854486677750085837425254720256 n 4 3 + 6125215851106554506656615262208 n + 8968610229255342387282781237248 n 2 + 8427526119144033265190253772800 n + 4520311863712029247503178137600 n + 1035468571266849540009885696000)/(59049 (3 n + 35) (3 n + 32) (3 n + 29) (3 n + 26) (3 n + 23) (3 n + 20) (3 n + 17) (3 n + 14) (3 n + 11) (3 n + 8) (3 n + 5) (3 n + 34) (3 n + 31) (3 n + 28) (3 n + 25) (3 n + 22) (3 n + 19) (3 n + 16) (3 n + 13) (3 n + 10) (3 n + 7) (3 n + 4) (n + 12) (n + 11) (n + 10) (n + 9) (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2)) B[12], = 195312500 ---------------- 1853020188851841 and in Maple notation 500/59049*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*(n-7)*(n-8)*(n-9)*(n-10)*(n-11)*( 390625*n^22+146093750*n^21+27263671875*n^20+3371893593750*n^19+308936639343750* n^18+22156447588027500*n^17+1279052684751943750*n^16+60115193382267897500*n^15+ 2299404607657354894125*n^14+70878598177548115189950*n^13+ 1729900820671159605447927*n^12+32666103637559910020696622*n^11+ 466201654576472678847544924*n^10+4950647702507197180824497504*n^9+ 38811779597851432703525662800*n^8+223575427080831090549810361440*n^7+ 940368325304867050278049974336*n^6+2854486677750085837425254720256*n^5+ 6125215851106554506656615262208*n^4+8968610229255342387282781237248*n^3+ 8427526119144033265190253772800*n^2+4520311863712029247503178137600*n+ 1035468571266849540009885696000)/(3*n+35)/(3*n+32)/(3*n+29)/(3*n+26)/(3*n+23)/( 3*n+20)/(3*n+17)/(3*n+14)/(3*n+11)/(3*n+8)/(3*n+5)/(3*n+34)/(3*n+31)/(3*n+28)/( 3*n+25)/(3*n+22)/(3*n+19)/(3*n+16)/(3*n+13)/(3*n+10)/(3*n+7)/(3*n+4)/(n+12)/(n+ 11)/(n+10)/(n+9)/(n+8)/(n+7)/(n+6)/(n+5)/(n+4)/(n+3)/(n+2) 195312500/1853020188851841 A[13, n], = 24 23 22 21 8125 (390625 n + 168750000 n + 36406093750 n + 5215711875000 n 20 19 18 + 555653435734375 n + 46629094852065000 n + 3180008817647860000 n 17 16 + 179018258758398972000 n + 8359337641821138259375 n 15 14 + 322668943357756914124800 n + 10190093136668578130540062 n 13 12 + 259128925919723685281352312 n + 5203152384672359378800158073 n 11 10 + 80911034292683730754499188968 n + 960859158382204140327863395372 n 9 8 + 8644816687803355258802624549616 n + 58639733337648209775385392345904 n 7 + 298365617966118297879376848036864 n 6 + 1129407850159607222809516005399360 n 5 + 3136439941072010200452070011439872 n 4 + 6245424914435425680218228384437248 n 3 + 8591178793062450532710004801671168 n 2 + 7666550139354642508656788490387456 n + 3942762690385205796234302521344000 n + 873521286720714271952339573145600 ) (n - 9) (n - 4) (n - 5) (n - 3) (n - 10) (n - 6) (n - 2) (n - 7) (n - 12) (n - 1) (n - 11) (n - 8)/(531441 (3 n + 37) (3 n + 38) (3 n + 13) (n + 5) (3 n + 14) (n + 11) (n + 13) (3 n + 17) (3 n + 16) (n + 6) (3 n + 22) (n + 8) (3 n + 34) (3 n + 19) (3 n + 35) (n + 12) (3 n + 29) (3 n + 25) (n + 9) (3 n + 26) (3 n + 11) (3 n + 10) (n + 4) (3 n + 5) (3 n + 4) (n + 2) (3 n + 31) (3 n + 23) (3 n + 28) (n + 10) (n + 7) (3 n + 20) (n + 3) (3 n + 8) (3 n + 7) (3 n + 32)) B[13], = 3173828125 ------------------ 150094635296999121 and in Maple notation 8125/531441/(3*n+37)/(3*n+38)*(390625*n^24+168750000*n^23+36406093750*n^22+ 5215711875000*n^21+555653435734375*n^20+46629094852065000*n^19+ 3180008817647860000*n^18+179018258758398972000*n^17+8359337641821138259375*n^16 +322668943357756914124800*n^15+10190093136668578130540062*n^14+ 259128925919723685281352312*n^13+5203152384672359378800158073*n^12+ 80911034292683730754499188968*n^11+960859158382204140327863395372*n^10+ 8644816687803355258802624549616*n^9+58639733337648209775385392345904*n^8+ 298365617966118297879376848036864*n^7+1129407850159607222809516005399360*n^6+ 3136439941072010200452070011439872*n^5+6245424914435425680218228384437248*n^4+ 8591178793062450532710004801671168*n^3+7666550139354642508656788490387456*n^2+ 3942762690385205796234302521344000*n+873521286720714271952339573145600)*(n-9)/( 3*n+13)/(n+5)/(3*n+14)*(n-4)/(n+11)/(n+13)/(3*n+17)/(3*n+16)/(n+6)/(3*n+22)/(n+ 8)*(n-5)/(3*n+34)*(n-3)*(n-10)*(n-6)/(3*n+19)*(n-2)/(3*n+35)/(n+12)/(3*n+29)*(n -7)/(3*n+25)/(n+9)/(3*n+26)/(3*n+11)/(3*n+10)/(n+4)*(n-12)/(3*n+5)/(3*n+4)/(n+2 )*(n-1)/(3*n+31)/(3*n+23)/(3*n+28)/(n+10)*(n-11)/(n+7)/(3*n+20)/(n+3)/(3*n+8)/( 3*n+7)/(3*n+32)*(n-8) 3173828125/150094635296999121 A[14, n], = 26 25 8750 (n - 13) (n - 9) (n - 4) (n - 5) (1953125 n + 964843750 n 24 23 22 + 238153906250 n + 39088696562500 n + 4783301683671875 n 21 20 + 463097850821856250 n + 36683217651310587500 n 19 18 + 2422106188714313530000 n + 134451183255111956250875 n 17 16 + 6280973764252578822252250 n + 245685651417340155574108010 n 15 14 + 7965254293912909544892495700 n + 210997650380295609314709054365 n 13 + 4490896894951729029703537650022 n 12 + 75553740331837563548537331395264 n 11 + 992245898107799045366426545042840 n 10 + 10092920710035456790344883467592160 n 9 + 79124024103314575984088177892139936 n 8 + 475943419824459295565069336112667392 n 7 + 2183035723828352120366849011221763200 n 6 + 7560042126451477031820410803218136320 n 5 + 19459298282268239851120990204607649792 n 4 + 36327240522566678668419897890954563584 n 3 + 47323256974226640625530005476733583360 n 2 + 40351431834497329990516280846482145280 n + 19989443234786436853023478605624115200 n + 4297724730665914218005510699876352000) (n - 3) (n - 10) (n - 6) (n - 2) (n - 7) (n - 12) (n - 1) (n - 11) (n - 8)/(1594323 (3 n + 37) (3 n + 38) (3 n + 41) (3 n + 40) (n + 14) (3 n + 13) (n + 5) (3 n + 14) (n + 11) (n + 13) (3 n + 17) (3 n + 16) (n + 6) (3 n + 22) (n + 8) (3 n + 34) (3 n + 19) (3 n + 35) (n + 12) (3 n + 29) (3 n + 25) (n + 9) (3 n + 26) (3 n + 11) (3 n + 10) (n + 4) (3 n + 5) (3 n + 4) (n + 2) (3 n + 31) (3 n + 23) (3 n + 28) (n + 10) (n + 7) (3 n + 20) (n + 3) (3 n + 8) (3 n + 7) (3 n + 32)) B[14], = 17089843750 ------------------- 4052555153018976267 and in Maple notation 8750/1594323/(3*n+37)/(3*n+38)*(n-13)/(3*n+41)/(3*n+40)/(n+14)*(n-9)/(3*n+13)/( n+5)/(3*n+14)*(n-4)/(n+11)/(n+13)/(3*n+17)/(3*n+16)/(n+6)/(3*n+22)/(n+8)*(n-5)* (1953125*n^26+964843750*n^25+238153906250*n^24+39088696562500*n^23+ 4783301683671875*n^22+463097850821856250*n^21+36683217651310587500*n^20+ 2422106188714313530000*n^19+134451183255111956250875*n^18+ 6280973764252578822252250*n^17+245685651417340155574108010*n^16+ 7965254293912909544892495700*n^15+210997650380295609314709054365*n^14+ 4490896894951729029703537650022*n^13+75553740331837563548537331395264*n^12+ 992245898107799045366426545042840*n^11+10092920710035456790344883467592160*n^10 +79124024103314575984088177892139936*n^9+475943419824459295565069336112667392*n ^8+2183035723828352120366849011221763200*n^7+ 7560042126451477031820410803218136320*n^6+ 19459298282268239851120990204607649792*n^5+ 36327240522566678668419897890954563584*n^4+ 47323256974226640625530005476733583360*n^3+ 40351431834497329990516280846482145280*n^2+ 19989443234786436853023478605624115200*n+4297724730665914218005510699876352000) /(3*n+34)*(n-3)*(n-10)*(n-6)/(3*n+19)*(n-2)/(3*n+35)/(n+12)/(3*n+29)*(n-7)/(3*n +25)/(n+9)/(3*n+26)/(3*n+11)/(3*n+10)/(n+4)*(n-12)/(3*n+5)/(3*n+4)/(n+2)*(n-1)/ (3*n+31)/(3*n+23)/(3*n+28)/(n+10)*(n-11)/(n+7)/(3*n+20)/(n+3)/(3*n+8)/(3*n+7)/( 3*n+32)*(n-8) 17089843750/4052555153018976267 A[15, n], = 3125 (n - 13) (n - 9) (n - 4) (n - 5) (n - 3) (n - 10) (n - 6) (n - 2) ( 28 27 26 25 9765625 n + 5468750000 n + 1530689453125 n + 285147931250000 n 24 23 + 39675651519765625 n + 4381244178311000000 n 22 21 + 397761085471676265625 n + 30313310875395097550000 n 20 19 + 1961174286891512081764375 n + 108173359307480208653912000 n 18 17 + 5080350687698668045856094175 n + 201968080938460004676743908400 n 16 + 6730476625682406293963951360275 n 15 + 185618071733653074203544652826240 n 14 + 4175519045287600533335553298262131 n 13 + 75547615625634842580331824774062672 n 12 + 1087185094408295711360414813396582116 n 11 + 12350540887794648498960848237810852432 n 10 + 110210195940320400685488505236041785488 n 9 + 769360198659614364958011391667398121856 n 8 + 4180702765066784579408128110506214560448 n 7 + 17554257240309290117180820322940103449856 n 6 + 56317168250050208409155683988784024339456 n 5 + 135716511520377820499131297787258639388672 n 4 + 239445513739029805942450641174325585575936 n 3 + 297275252835249599670681722258134702620672 n 2 + 243407207633160475975692129041365519564800 n + 116590431719022121929785188974875954380800 n + 24393885571259729101399278732498173952000) (n - 7) (n - 14) (n - 12) (n - 1) (n - 11) (n - 8)/(1594323 (3 n + 37) (3 n + 38) (3 n + 41) (3 n + 44) (3 n + 43) (3 n + 40) (n + 14) (n + 15) (3 n + 13) (n + 5) (3 n + 14) (n + 11) (n + 13) (3 n + 17) (3 n + 16) (n + 6) (3 n + 22) (n + 8) (3 n + 34) (3 n + 19) (3 n + 35) (n + 12) (3 n + 29) (3 n + 25) (n + 9) (3 n + 26) (3 n + 11) (3 n + 10) (n + 4) (3 n + 5) (3 n + 4) (n + 2) (3 n + 31) (3 n + 23) (3 n + 28) (n + 10) (n + 7) (3 n + 20) (n + 3) (3 n + 8) (3 n + 7) (3 n + 32)) B[15], = 30517578125 -------------------- 36472996377170786403 and in Maple notation 3125/1594323/(3*n+37)/(3*n+38)*(n-13)/(3*n+41)/(3*n+44)/(3*n+43)/(3*n+40)/(n+14 )/(n+15)*(n-9)/(3*n+13)/(n+5)/(3*n+14)*(n-4)/(n+11)/(n+13)/(3*n+17)/(3*n+16)/(n +6)/(3*n+22)/(n+8)*(n-5)/(3*n+34)*(n-3)*(n-10)*(n-6)/(3*n+19)*(n-2)/(3*n+35)*( 9765625*n^28+5468750000*n^27+1530689453125*n^26+285147931250000*n^25+ 39675651519765625*n^24+4381244178311000000*n^23+397761085471676265625*n^22+ 30313310875395097550000*n^21+1961174286891512081764375*n^20+ 108173359307480208653912000*n^19+5080350687698668045856094175*n^18+ 201968080938460004676743908400*n^17+6730476625682406293963951360275*n^16+ 185618071733653074203544652826240*n^15+4175519045287600533335553298262131*n^14+ 75547615625634842580331824774062672*n^13+1087185094408295711360414813396582116* n^12+12350540887794648498960848237810852432*n^11+ 110210195940320400685488505236041785488*n^10+ 769360198659614364958011391667398121856*n^9+ 4180702765066784579408128110506214560448*n^8+ 17554257240309290117180820322940103449856*n^7+ 56317168250050208409155683988784024339456*n^6+ 135716511520377820499131297787258639388672*n^5+ 239445513739029805942450641174325585575936*n^4+ 297275252835249599670681722258134702620672*n^3+ 243407207633160475975692129041365519564800*n^2+ 116590431719022121929785188974875954380800*n+ 24393885571259729101399278732498173952000)/(n+12)/(3*n+29)*(n-7)*(n-14)/(3*n+25 )/(n+9)/(3*n+26)/(3*n+11)/(3*n+10)/(n+4)*(n-12)/(3*n+5)/(3*n+4)/(n+2)*(n-1)/(3* n+31)/(3*n+23)/(3*n+28)/(n+10)*(n-11)/(n+7)/(3*n+20)/(n+3)/(3*n+8)/(3*n+7)/(3*n +32)*(n-8) 30517578125/36472996377170786403 A[16, n], = 10000 (n - 13) (n - 9) (n - 4) (n - 5) (n - 15) (n - 3) (n - 10) (n - 6) 30 29 28 (n - 2) (48828125 n + 30761718750 n + 9688134765625 n 27 26 25 + 2031960410156250 n + 318722301228515625 n + 39764670462178593750 n 24 23 + 4093182394572664453125 n + 355509559005491273906250 n 22 21 + 26401508061638197990734375 n + 1687675729508032133300681250 n 20 19 + 93004543600428939236104858875 n + 4406778516261193679627720358750 n 18 + 178418590210040512322228554921875 n 17 + 6115907790201190950611044547465250 n 16 + 175467032242094490797075924980633575 n 15 + 4160541182889464276961439393131583134 n 14 + 80543625256002873677866111978962969600 n 13 + 1260368836589197634478427098020727091080 n 12 + 15828782284181089257133586730516581719840 n 11 + 158766249082365277105871429046512177008608 n 10 + 1266907447234675719929487284302534765679360 n 9 + 8009107483063817177214299882402164796269440 n 8 + 39883271429297735241457997296641364094027520 n 7 + 155139658233010354689023083530368301118405632 n 6 + 465613241951116716389626897049245659160289280 n 5 + 1058930175220137984397904168758941951561646080 n 4 + 1777083076464111693872158852837728810527293440 n 3 + 2113538786375237861756588628681787508248805376 n 2 + 1668577372688137461847997715434996419012853760 n + 775237523777093068874655642318190772905574400 n + 158218741815190602951675721858983156252672000) (n - 7) (n - 14) (n - 12) (n - 1) (n - 11) (n - 8)/(14348907 (3 n + 37) (3 n + 38) (3 n + 47) (3 n + 41) (3 n + 44) (3 n + 43) (3 n + 40) (n + 14) (n + 15) (n + 16) (3 n + 13) (n + 5) (3 n + 14) (n + 11) (n + 13) (3 n + 17) (3 n + 16) (n + 6) (3 n + 22) (n + 8) (3 n + 34) (3 n + 46) (3 n + 19) (3 n + 35) (n + 12) (3 n + 29) (3 n + 25) (n + 9) (3 n + 26) (3 n + 11) (3 n + 10) (n + 4) (3 n + 5) (3 n + 4) (n + 2) (3 n + 31) (3 n + 23) (3 n + 28) (n + 10) (n + 7) (3 n + 20) (n + 3) (3 n + 8) (3 n + 7) (3 n + 32)) B[16], = 488281250000 ---------------------- 2954312706550833698643 and in Maple notation 10000/14348907/(3*n+37)/(3*n+38)*(n-13)/(3*n+47)/(3*n+41)/(3*n+44)/(3*n+43)/(3* n+40)/(n+14)/(n+15)*(n-9)/(n+16)/(3*n+13)/(n+5)/(3*n+14)*(n-4)/(n+11)/(n+13)/(3 *n+17)/(3*n+16)/(n+6)/(3*n+22)/(n+8)*(n-5)*(n-15)/(3*n+34)*(n-3)*(n-10)/(3*n+46 )*(n-6)/(3*n+19)*(n-2)*(48828125*n^30+30761718750*n^29+9688134765625*n^28+ 2031960410156250*n^27+318722301228515625*n^26+39764670462178593750*n^25+ 4093182394572664453125*n^24+355509559005491273906250*n^23+ 26401508061638197990734375*n^22+1687675729508032133300681250*n^21+ 93004543600428939236104858875*n^20+4406778516261193679627720358750*n^19+ 178418590210040512322228554921875*n^18+6115907790201190950611044547465250*n^17+ 175467032242094490797075924980633575*n^16+4160541182889464276961439393131583134 *n^15+80543625256002873677866111978962969600*n^14+ 1260368836589197634478427098020727091080*n^13+ 15828782284181089257133586730516581719840*n^12+ 158766249082365277105871429046512177008608*n^11+ 1266907447234675719929487284302534765679360*n^10+ 8009107483063817177214299882402164796269440*n^9+ 39883271429297735241457997296641364094027520*n^8+ 155139658233010354689023083530368301118405632*n^7+ 465613241951116716389626897049245659160289280*n^6+ 1058930175220137984397904168758941951561646080*n^5+ 1777083076464111693872158852837728810527293440*n^4+ 2113538786375237861756588628681787508248805376*n^3+ 1668577372688137461847997715434996419012853760*n^2+ 775237523777093068874655642318190772905574400*n+ 158218741815190602951675721858983156252672000)/(3*n+35)/(n+12)/(3*n+29)*(n-7)*( n-14)/(3*n+25)/(n+9)/(3*n+26)/(3*n+11)/(3*n+10)/(n+4)*(n-12)/(3*n+5)/(3*n+4)/(n +2)*(n-1)/(3*n+31)/(3*n+23)/(3*n+28)/(n+10)*(n-11)/(n+7)/(3*n+20)/(n+3)/(3*n+8) /(3*n+7)/(3*n+32)*(n-8) 488281250000/2954312706550833698643 A[17, n], = 32 31 30 53125 (48828125 n + 34375000000 n + 12098828125000 n 29 28 27 + 2837015937500000 n + 497954212054687500 n + 69631182156532500000 n 26 25 + 8054121580992890625000 n + 789081904306868932500000 n 24 23 + 66457407260279202073968750 n + 4852600418265527436143700000 n 22 + 308331627128829839623809357000 n 21 + 17044594867995638721547301532000 n 20 + 816904878627420459711031509451500 n 19 + 33732480108678267613779558115932000 n 18 + 1189817999730359074000536545177088600 n 17 + 35480116109810396395927364569743253344 n 16 + 884559043914259587860093537800612890333 n 15 + 18241550264954551682697722257424720538528 n 14 + 308380288387426685676777556150201768331824 n 13 + 4245078629321314054572690168051328315823552 n 12 + 47357072559296979825413464140113357324536160 n 11 + 426547035911857122446455778738594295051708160 n 10 + 3090489064358021998406449857791159047548840192 n 9 + 17928905220873526226783376549062544302328689664 n 8 + 82744113328844656699834210313757644321233126656 n 7 + 300997423230385027747162712890582600999433220096 n 6 + 851738025635381162513359098560364285746019999744 n 5 + 1839902625040310549552774986957893370621109207040 n 4 + 2952482469677132119736370482567337249575897726976 n 3 + 3378244598858349225947722186059755141833310601216 n 2 + 2580284135802053488817923687968209123871440240640 n + 1165929436813806274671864855622038676023174758400 n + 232581550468330186338963311132705239691427840000) (n - 13) (n - 9) (n - 16) (n - 4) (n - 5) (n - 15) (n - 3) (n - 10) (n - 6) (n - 2) (n - 7) (n - 14) (n - 12) (n - 1) (n - 11) (n - 8)/(43046721 (3 n + 37) (3 n + 38) (3 n + 50) (3 n + 47) (3 n + 41) (3 n + 44) (3 n + 43) (3 n + 40) (n + 14) (3 n + 49) (n + 15) (n + 16) (3 n + 13) (n + 5) (3 n + 14) (n + 11) (n + 13) (3 n + 17) (3 n + 16) (n + 6) (3 n + 22) (n + 8) (3 n + 34) (3 n + 46) (3 n + 19) (n + 17) (3 n + 35) (n + 12) (3 n + 29) (3 n + 25) (n + 9) (3 n + 26) (3 n + 11) (3 n + 10) (n + 4) (3 n + 5) (3 n + 4) (n + 2) (3 n + 31) (3 n + 23) (3 n + 28) (n + 10) (n + 7) (3 n + 20) (n + 3) (3 n + 8) (3 n + 7) (3 n + 32)) B[17], = 2593994140625 ----------------------- 79766443076872509863361 and in Maple notation 53125/43046721/(3*n+37)/(3*n+38)*(48828125*n^32+34375000000*n^31+12098828125000 *n^30+2837015937500000*n^29+497954212054687500*n^28+69631182156532500000*n^27+ 8054121580992890625000*n^26+789081904306868932500000*n^25+ 66457407260279202073968750*n^24+4852600418265527436143700000*n^23+ 308331627128829839623809357000*n^22+17044594867995638721547301532000*n^21+ 816904878627420459711031509451500*n^20+33732480108678267613779558115932000*n^19 +1189817999730359074000536545177088600*n^18+ 35480116109810396395927364569743253344*n^17+ 884559043914259587860093537800612890333*n^16+ 18241550264954551682697722257424720538528*n^15+ 308380288387426685676777556150201768331824*n^14+ 4245078629321314054572690168051328315823552*n^13+ 47357072559296979825413464140113357324536160*n^12+ 426547035911857122446455778738594295051708160*n^11+ 3090489064358021998406449857791159047548840192*n^10+ 17928905220873526226783376549062544302328689664*n^9+ 82744113328844656699834210313757644321233126656*n^8+ 300997423230385027747162712890582600999433220096*n^7+ 851738025635381162513359098560364285746019999744*n^6+ 1839902625040310549552774986957893370621109207040*n^5+ 2952482469677132119736370482567337249575897726976*n^4+ 3378244598858349225947722186059755141833310601216*n^3+ 2580284135802053488817923687968209123871440240640*n^2+ 1165929436813806274671864855622038676023174758400*n+ 232581550468330186338963311132705239691427840000)/(3*n+50)*(n-13)/(3*n+47)/(3*n +41)/(3*n+44)/(3*n+43)/(3*n+40)/(n+14)/(3*n+49)/(n+15)*(n-9)*(n-16)/(n+16)/(3*n +13)/(n+5)/(3*n+14)*(n-4)/(n+11)/(n+13)/(3*n+17)/(3*n+16)/(n+6)/(3*n+22)/(n+8)* (n-5)*(n-15)/(3*n+34)*(n-3)*(n-10)/(3*n+46)*(n-6)/(3*n+19)*(n-2)/(n+17)/(3*n+35 )/(n+12)/(3*n+29)*(n-7)*(n-14)/(3*n+25)/(n+9)/(3*n+26)/(3*n+11)/(3*n+10)/(n+4)* (n-12)/(3*n+5)/(3*n+4)/(n+2)*(n-1)/(3*n+31)/(3*n+23)/(3*n+28)/(n+10)*(n-11)/(n+ 7)/(3*n+20)/(n+3)/(3*n+8)/(3*n+7)/(3*n+32)*(n-8) 2593994140625/79766443076872509863361 A[18, n], = 34 33 6250 (n - 13) (n - 9) (n - 16) (244140625 n + 190917968750 n 32 31 30 + 74643945312500 n + 19447827343750000 n + 3795140769023437500 n 29 28 + 590715588712678125000 n + 76200620497089196875000 n 27 26 + 8349768557831098323750000 n + 789714699761757100359843750 n 25 24 + 65111271074187596547731812500 n + 4704963199770709857745896210000 n 23 + 298475557300670817686093213310000 n 22 + 16600928559745826773109534636977500 n 21 + 806339106594780516257949904858485000 n 20 + 33992919733558248797332528579647777000 n 19 + 1233878196807261142616018656964638287120 n 18 + 38205643879238229696493076840954420200785 n 17 + 999233495284579781182409445153208426006254 n 16 + 21866478728478728658545291561403798476729564 n 15 + 397136883066772240002393783158541696296369632 n 14 + 5948719235874821227042049982082909543604254240 n 13 + 73150962515938950006943221689430537586881351488 n 12 + 735814882876748478024018536155583061151458553728 n 11 + 6034093881412129568061181303199971047165018868224 n 10 + 40184656718884941771650879675282816494753458504960 n 9 + 216209429325373165323638533542645311590469549747712 n 8 + 933157011089514333111211786701816648709387998538752 n 7 + 3198743737968178034888385872733837216996331630411776 n 6 + 8588738222653626476772260764753295187340301147013120 n 5 + 17715930099579986138191473897119920969270288094724096 n 4 + 27302701923524841230211676337005868780809103186067456 n 3 + 30162097806254360624178128457752657158289245756981248 n 2 + 22352659191727591595213853500879628620622263243243520 n + 9845515372852298275454272112110124347506442842931200 n + 1922984259272153980650548656445206921768725381120000) (n - 4) (n - 5) (n - 15) (n - 3) (n - 10) (n - 6) (n - 2) (n - 17) (n - 7) (n - 14) (n - 12) (n - 1) (n - 11) (n - 8)/(14348907 (3 n + 37) (3 n + 38) (3 n + 52) (3 n + 50) (3 n + 47) (n + 18) (3 n + 41) (3 n + 44) (3 n + 43) (3 n + 40) (n + 14) (3 n + 49) (n + 15) (n + 16) (3 n + 13) (n + 5) (3 n + 14) (n + 11) (n + 13) (3 n + 17) (3 n + 16) (n + 6) (3 n + 22) (n + 8) (3 n + 34) (3 n + 46) (3 n + 19) (n + 17) (3 n + 35) (n + 12) (3 n + 29) (3 n + 25) (n + 9) (3 n + 26) (3 n + 11) (3 n + 10) (n + 4) (3 n + 5) (3 n + 4) (n + 2) (3 n + 31) (3 n + 23) (3 n + 28) (n + 10) (3 n + 53) (n + 7) (3 n + 20) (n + 3) (3 n + 8) (3 n + 7) (3 n + 32)) B[18], = 1525878906250 ------------------------ 239299329230617529590083 and in Maple notation 6250/14348907/(3*n+37)/(3*n+38)/(3*n+52)/(3*n+50)*(n-13)/(3*n+47)/(n+18)/(3*n+ 41)/(3*n+44)/(3*n+43)/(3*n+40)/(n+14)/(3*n+49)/(n+15)*(n-9)*(n-16)/(n+16)*( 244140625*n^34+190917968750*n^33+74643945312500*n^32+19447827343750000*n^31+ 3795140769023437500*n^30+590715588712678125000*n^29+76200620497089196875000*n^ 28+8349768557831098323750000*n^27+789714699761757100359843750*n^26+ 65111271074187596547731812500*n^25+4704963199770709857745896210000*n^24+ 298475557300670817686093213310000*n^23+16600928559745826773109534636977500*n^22 +806339106594780516257949904858485000*n^21+ 33992919733558248797332528579647777000*n^20+ 1233878196807261142616018656964638287120*n^19+ 38205643879238229696493076840954420200785*n^18+ 999233495284579781182409445153208426006254*n^17+ 21866478728478728658545291561403798476729564*n^16+ 397136883066772240002393783158541696296369632*n^15+ 5948719235874821227042049982082909543604254240*n^14+ 73150962515938950006943221689430537586881351488*n^13+ 735814882876748478024018536155583061151458553728*n^12+ 6034093881412129568061181303199971047165018868224*n^11+ 40184656718884941771650879675282816494753458504960*n^10+ 216209429325373165323638533542645311590469549747712*n^9+ 933157011089514333111211786701816648709387998538752*n^8+ 3198743737968178034888385872733837216996331630411776*n^7+ 8588738222653626476772260764753295187340301147013120*n^6+ 17715930099579986138191473897119920969270288094724096*n^5+ 27302701923524841230211676337005868780809103186067456*n^4+ 30162097806254360624178128457752657158289245756981248*n^3+ 22352659191727591595213853500879628620622263243243520*n^2+ 9845515372852298275454272112110124347506442842931200*n+ 1922984259272153980650548656445206921768725381120000)/(3*n+13)/(n+5)/(3*n+14)*( n-4)/(n+11)/(n+13)/(3*n+17)/(3*n+16)/(n+6)/(3*n+22)/(n+8)*(n-5)*(n-15)/(3*n+34) *(n-3)*(n-10)/(3*n+46)*(n-6)/(3*n+19)*(n-2)/(n+17)/(3*n+35)/(n+12)/(3*n+29)*(n-\ 17)*(n-7)*(n-14)/(3*n+25)/(n+9)/(3*n+26)/(3*n+11)/(3*n+10)/(n+4)*(n-12)/(3*n+5) /(3*n+4)/(n+2)*(n-1)/(3*n+31)/(3*n+23)/(3*n+28)/(n+10)*(n-11)/(3*n+53)/(n+7)/(3 *n+20)/(n+3)/(3*n+8)/(3*n+7)/(3*n+32)*(n-8) 1525878906250/239299329230617529590083 To sum up, here are the, A[r, n], for f from 2 to , 18 2 10 (n - 1) (n + 14 n + 12) [-----------------------------, 3 (3 n + 5) (3 n + 4) (n + 2) 4 3 2 5 (n - 1) (n - 2) (5 n + 160 n + 1803 n + 3768 n + 2016) -----------------------------------------------------------, 20 (n - 1) 3 (3 n + 8) (3 n + 5) (3 n + 7) (3 n + 4) (n + 3) (n + 2) (n - 2) (n - 3) 6 5 4 3 2 (25 n + 1350 n + 31495 n + 347406 n + 1211092 n + 1580304 n + 665280)/ (27 (3 n + 11) (3 n + 8) (3 n + 5) (3 n + 10) (3 n + 7) (3 n + 4) (n + 4) 8 7 (n + 3) (n + 2)), 25 (n - 1) (n - 2) (n - 3) (n - 4) (125 n + 10000 n 6 5 4 3 2 + 371450 n + 7831720 n + 90056453 n + 442852216 n + 996421764 n + 1007115984 n + 363242880)/(81 (3 n + 14) (3 n + 11) (3 n + 8) (3 n + 5) (3 n + 13) (3 n + 10) (3 n + 7) (3 n + 4) (n + 5) (n + 4) (n + 3) (n + 2)), 10 9 8 10 (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (625 n + 68750 n + 3628750 n 7 6 5 4 + 116901500 n + 2395396825 n + 29421692462 n + 186231365400 n 3 2 + 617582906760 n + 1076423055120 n + 918818062848 n + 296406190080)/(81 (3 n + 17) (3 n + 14) (3 n + 11) (3 n + 8) (3 n + 5) (3 n + 16) (3 n + 13) (3 n + 10) (3 n + 7) (3 n + 4) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2)), 12 11 175 (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (625 n + 90000 n 10 9 8 7 + 6325625 n + 281268000 n + 8516307975 n + 175323137472 n 6 5 4 + 2320161434435 n + 17886472474464 n + 79792191915340 n 3 2 + 206727295234608 n + 302645014842240 n + 228319317305856 n + 67580611338240)/(729 (3 n + 20) (3 n + 17) (3 n + 14) (3 n + 11) (3 n + 8) (3 n + 5) (3 n + 19) (3 n + 16) (3 n + 13) (3 n + 10) (3 n + 7) (3 n + 4) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2)), 200 (n - 1) 14 13 (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (n - 7) (3125 n + 568750 n 12 11 10 9 + 51008125 n + 2952031250 n + 120616941375 n + 3564549572010 n 8 7 6 + 75011757331015 n + 1071261463254878 n + 9703709965152728 n 5 4 3 + 54870097208217592 n + 193681878688205040 n + 421496433468194112 n 2 + 541984884627439872 n + 371487057175922688 n + 102587368011448320)/(2187 (3 n + 23) (3 n + 20) (3 n + 17) (3 n + 14) (3 n + 11) (3 n + 8) (3 n + 5) (3 n + 22) (3 n + 19) (3 n + 16) (3 n + 13) (3 n + 10) (3 n + 7) (3 n + 4) (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2)), 25 (n - 1) 16 (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (n - 7) (n - 8) (15625 n 15 14 13 12 + 3500000 n + 388512500 n + 28157710000 n + 1471455483750 n 11 10 9 + 57589291846800 n + 1695686514421100 n + 36823499062435120 n 8 7 6 + 566623013737989961 n + 5883512920147660112 n + 40461585694240387992 n 5 4 + 183663017691739672800 n + 546422776901073190224 n 3 2 + 1041865968637322656128 n + 1209989486787870567168 n + 767428785578256721920 n + 200045367622324224000)/(729 (3 n + 26) (3 n + 23) (3 n + 20) (3 n + 17) (3 n + 14) (3 n + 11) (3 n + 8) (3 n + 5) (3 n + 25) (3 n + 22) (3 n + 19) (3 n + 16) (3 n + 13) (3 n + 10) (3 n + 7) (3 n + 4) (n + 9) (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2)), 250 (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (n - 7) (n - 8) (n - 9) 18 17 16 15 (78125 n + 21093750 n + 2832000000 n + 250057575000 n 14 13 12 + 16129151298750 n + 795938987686500 n + 30534485772468500 n 11 10 + 906798973422904200 n + 20427547655336548845 n 9 8 + 337686950884283521398 n + 3947955480810255108516 n 7 6 + 32043500652066554061936 n + 179512767333384682604720 n 5 4 + 690539558794947366099552 n + 1800054096927431745057984 n 3 2 + 3089887924189906537726464 n + 3302805644601397253544960 n + 1963602485859499681996800 n + 487310515527981809664000)/(19683 (3 n + 29) (3 n + 26) (3 n + 23) (3 n + 20) (3 n + 17) (3 n + 14) (3 n + 11) (3 n + 8) (3 n + 5) (3 n + 28) (3 n + 25) (3 n + 22) (3 n + 19) (3 n + 16) (3 n + 13) (3 n + 10) (3 n + 7) (3 n + 4) (n + 10) (n + 9) (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2)), 275 (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (n - 7) (n - 8) (n - 9) (n - 10) ( 20 19 18 17 390625 n + 125000000 n + 19932421875 n + 2100058125000 n 16 15 14 + 162998007281250 n + 9812713709640000 n + 468929848263193750 n 13 12 + 17882056172995670000 n + 539717687867022031125 n 11 10 + 12647249945053650458400 n + 223840869361781977857207 n 9 8 + 2906822815545027467301000 n + 27223159945086515727267640 n 7 6 + 182549042700526995526230560 n + 872351065103429658321105456 n 5 4 + 2945228241209805686343502080 n + 6902569938133543227991752960 n 3 2 + 10868899678282135763981015040 n + 10838203759365349592131264512 n + 6098034294882387606412984320 n + 1450236094211273865560064000)/(59049 (3 n + 32) (3 n + 29) (3 n + 26) (3 n + 23) (3 n + 20) (3 n + 17) (3 n + 14) (3 n + 11) (3 n + 8) (3 n + 5) (3 n + 31) (3 n + 28) (3 n + 25) (3 n + 22) (3 n + 19) (3 n + 16) (3 n + 13) (3 n + 10) (3 n + 7) (3 n + 4) (n + 11) (n + 10) (n + 9) (n + 8) (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2)), 500 (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) (n - 6) (n - 7) 22 21 (n - 8) (n - 9) (n - 10) (n - 11) (390625 n + 146093750 n 20 19 18 + 27263671875 n + 3371893593750 n + 308936639343750 n 17 16 + 22156447588027500 n + 1279052684751943750 n 15 14 + 60115193382267897500 n + 2299404607657354894125 n 13 12 + 70878598177548115189950 n + 1729900820671159605447927 n 11 10 + 32666103637559910020696622 n + 466201654576472678847544924 n 9 8 + 4950647702507197180824497504 n + 38811779597851432703525662800 n 7 6 + 223575427080831090549810361440 n + 940368325304867050278049974336 n 5 4 + 2854486677750085837425254720256 n + 6125215851106554506656615262208 n 3 2 + 8968610229255342387282781237248 n + 8427526119144033265190253772800 n + 4520311863712029247503178137600 n + 1035468571266849540009885696000)/( 59049 (3 n + 35) (3 n + 32) (3 n + 29) (3 n + 26) (3 n + 23) (3 n + 20) (3 n + 17) (3 n + 14) (3 n + 11) (3 n + 8) (3 n + 5) (3 n + 34) (3 n + 31) (3 n + 28) (3 n + 25) (3 n + 22) (3 n + 19) (3 n + 16) (3 n + 13) (3 n + 10) (3 n + 7) (3 n + 4) (n + 12) (n + 11) (n + 10) (n + 9) (n + 8) 24 (n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2)), 8125 (390625 n 23 22 21 + 168750000 n + 36406093750 n + 5215711875000 n 20 19 18 + 555653435734375 n + 46629094852065000 n + 3180008817647860000 n 17 16 + 179018258758398972000 n + 8359337641821138259375 n 15 14 + 322668943357756914124800 n + 10190093136668578130540062 n 13 12 + 259128925919723685281352312 n + 5203152384672359378800158073 n 11 10 + 80911034292683730754499188968 n + 960859158382204140327863395372 n 9 8 + 8644816687803355258802624549616 n + 58639733337648209775385392345904 n 7 + 298365617966118297879376848036864 n 6 + 1129407850159607222809516005399360 n 5 + 3136439941072010200452070011439872 n 4 + 6245424914435425680218228384437248 n 3 + 8591178793062450532710004801671168 n 2 + 7666550139354642508656788490387456 n + 3942762690385205796234302521344000 n + 873521286720714271952339573145600 ) (n - 9) (n - 4) (n - 5) (n - 3) (n - 10) (n - 6) (n - 2) (n - 7) (n - 12) (n - 1) (n - 11) (n - 8)/(531441 (3 n + 37) (3 n + 38) (3 n + 13) (n + 5) (3 n + 14) (n + 11) (n + 13) (3 n + 17) (3 n + 16) (n + 6) (3 n + 22) (n + 8) (3 n + 34) (3 n + 19) (3 n + 35) (n + 12) (3 n + 29) (3 n + 25) (n + 9) (3 n + 26) (3 n + 11) (3 n + 10) (n + 4) (3 n + 5) (3 n + 4) (n + 2) (3 n + 31) (3 n + 23) (3 n + 28) (n + 10) (n + 7) (3 n + 20) (n + 3) (3 n + 8) (3 n + 7) (3 n + 32)), 8750 (n - 13) (n - 9) (n - 4) 26 25 24 (n - 5) (1953125 n + 964843750 n + 238153906250 n 23 22 21 + 39088696562500 n + 4783301683671875 n + 463097850821856250 n 20 19 + 36683217651310587500 n + 2422106188714313530000 n 18 17 + 134451183255111956250875 n + 6280973764252578822252250 n 16 15 + 245685651417340155574108010 n + 7965254293912909544892495700 n 14 13 + 210997650380295609314709054365 n + 4490896894951729029703537650022 n 12 + 75553740331837563548537331395264 n 11 + 992245898107799045366426545042840 n 10 + 10092920710035456790344883467592160 n 9 + 79124024103314575984088177892139936 n 8 + 475943419824459295565069336112667392 n 7 + 2183035723828352120366849011221763200 n 6 + 7560042126451477031820410803218136320 n 5 + 19459298282268239851120990204607649792 n 4 + 36327240522566678668419897890954563584 n 3 + 47323256974226640625530005476733583360 n 2 + 40351431834497329990516280846482145280 n + 19989443234786436853023478605624115200 n + 4297724730665914218005510699876352000) (n - 3) (n - 10) (n - 6) (n - 2) (n - 7) (n - 12) (n - 1) (n - 11) (n - 8)/(1594323 (3 n + 37) (3 n + 38) (3 n + 41) (3 n + 40) (n + 14) (3 n + 13) (n + 5) (3 n + 14) (n + 11) (n + 13) (3 n + 17) (3 n + 16) (n + 6) (3 n + 22) (n + 8) (3 n + 34) (3 n + 19) (3 n + 35) (n + 12) (3 n + 29) (3 n + 25) (n + 9) (3 n + 26) (3 n + 11) (3 n + 10) (n + 4) (3 n + 5) (3 n + 4) (n + 2) (3 n + 31) (3 n + 23) (3 n + 28) (n + 10) (n + 7) (3 n + 20) (n + 3) (3 n + 8) (3 n + 7) (3 n + 32)), 3125 (n - 13) (n - 9) (n - 4) (n - 5) (n - 3) 28 27 26 (n - 10) (n - 6) (n - 2) (9765625 n + 5468750000 n + 1530689453125 n 25 24 23 + 285147931250000 n + 39675651519765625 n + 4381244178311000000 n 22 21 + 397761085471676265625 n + 30313310875395097550000 n 20 19 + 1961174286891512081764375 n + 108173359307480208653912000 n 18 17 + 5080350687698668045856094175 n + 201968080938460004676743908400 n 16 + 6730476625682406293963951360275 n 15 + 185618071733653074203544652826240 n 14 + 4175519045287600533335553298262131 n 13 + 75547615625634842580331824774062672 n 12 + 1087185094408295711360414813396582116 n 11 + 12350540887794648498960848237810852432 n 10 + 110210195940320400685488505236041785488 n 9 + 769360198659614364958011391667398121856 n 8 + 4180702765066784579408128110506214560448 n 7 + 17554257240309290117180820322940103449856 n 6 + 56317168250050208409155683988784024339456 n 5 + 135716511520377820499131297787258639388672 n 4 + 239445513739029805942450641174325585575936 n 3 + 297275252835249599670681722258134702620672 n 2 + 243407207633160475975692129041365519564800 n + 116590431719022121929785188974875954380800 n + 24393885571259729101399278732498173952000) (n - 7) (n - 14) (n - 12) (n - 1) (n - 11) (n - 8)/(1594323 (3 n + 37) (3 n + 38) (3 n + 41) (3 n + 44) (3 n + 43) (3 n + 40) (n + 14) (n + 15) (3 n + 13) (n + 5) (3 n + 14) (n + 11) (n + 13) (3 n + 17) (3 n + 16) (n + 6) (3 n + 22) (n + 8) (3 n + 34) (3 n + 19) (3 n + 35) (n + 12) (3 n + 29) (3 n + 25) (n + 9) (3 n + 26) (3 n + 11) (3 n + 10) (n + 4) (3 n + 5) (3 n + 4) (n + 2) (3 n + 31) (3 n + 23) (3 n + 28) (n + 10) (n + 7) (3 n + 20) (n + 3) (3 n + 8) (3 n + 7) (3 n + 32)), 10000 (n - 13) (n - 9) (n - 4) 30 (n - 5) (n - 15) (n - 3) (n - 10) (n - 6) (n - 2) (48828125 n 29 28 27 + 30761718750 n + 9688134765625 n + 2031960410156250 n 26 25 + 318722301228515625 n + 39764670462178593750 n 24 23 + 4093182394572664453125 n + 355509559005491273906250 n 22 21 + 26401508061638197990734375 n + 1687675729508032133300681250 n 20 19 + 93004543600428939236104858875 n + 4406778516261193679627720358750 n 18 + 178418590210040512322228554921875 n 17 + 6115907790201190950611044547465250 n 16 + 175467032242094490797075924980633575 n 15 + 4160541182889464276961439393131583134 n 14 + 80543625256002873677866111978962969600 n 13 + 1260368836589197634478427098020727091080 n 12 + 15828782284181089257133586730516581719840 n 11 + 158766249082365277105871429046512177008608 n 10 + 1266907447234675719929487284302534765679360 n 9 + 8009107483063817177214299882402164796269440 n 8 + 39883271429297735241457997296641364094027520 n 7 + 155139658233010354689023083530368301118405632 n 6 + 465613241951116716389626897049245659160289280 n 5 + 1058930175220137984397904168758941951561646080 n 4 + 1777083076464111693872158852837728810527293440 n 3 + 2113538786375237861756588628681787508248805376 n 2 + 1668577372688137461847997715434996419012853760 n + 775237523777093068874655642318190772905574400 n + 158218741815190602951675721858983156252672000) (n - 7) (n - 14) (n - 12) (n - 1) (n - 11) (n - 8)/(14348907 (3 n + 37) (3 n + 38) (3 n + 47) (3 n + 41) (3 n + 44) (3 n + 43) (3 n + 40) (n + 14) (n + 15) (n + 16) (3 n + 13) (n + 5) (3 n + 14) (n + 11) (n + 13) (3 n + 17) (3 n + 16) (n + 6) (3 n + 22) (n + 8) (3 n + 34) (3 n + 46) (3 n + 19) (3 n + 35) (n + 12) (3 n + 29) (3 n + 25) (n + 9) (3 n + 26) (3 n + 11) (3 n + 10) (n + 4) (3 n + 5) (3 n + 4) (n + 2) (3 n + 31) (3 n + 23) (3 n + 28) (n + 10) (n + 7) (3 n + 20) (n + 3) (3 n + 8) (3 n + 7) (3 n + 32)), 53125 32 31 30 29 (48828125 n + 34375000000 n + 12098828125000 n + 2837015937500000 n 28 27 + 497954212054687500 n + 69631182156532500000 n 26 25 + 8054121580992890625000 n + 789081904306868932500000 n 24 23 + 66457407260279202073968750 n + 4852600418265527436143700000 n 22 + 308331627128829839623809357000 n 21 + 17044594867995638721547301532000 n 20 + 816904878627420459711031509451500 n 19 + 33732480108678267613779558115932000 n 18 + 1189817999730359074000536545177088600 n 17 + 35480116109810396395927364569743253344 n 16 + 884559043914259587860093537800612890333 n 15 + 18241550264954551682697722257424720538528 n 14 + 308380288387426685676777556150201768331824 n 13 + 4245078629321314054572690168051328315823552 n 12 + 47357072559296979825413464140113357324536160 n 11 + 426547035911857122446455778738594295051708160 n 10 + 3090489064358021998406449857791159047548840192 n 9 + 17928905220873526226783376549062544302328689664 n 8 + 82744113328844656699834210313757644321233126656 n 7 + 300997423230385027747162712890582600999433220096 n 6 + 851738025635381162513359098560364285746019999744 n 5 + 1839902625040310549552774986957893370621109207040 n 4 + 2952482469677132119736370482567337249575897726976 n 3 + 3378244598858349225947722186059755141833310601216 n 2 + 2580284135802053488817923687968209123871440240640 n + 1165929436813806274671864855622038676023174758400 n + 232581550468330186338963311132705239691427840000) (n - 13) (n - 9) (n - 16) (n - 4) (n - 5) (n - 15) (n - 3) (n - 10) (n - 6) (n - 2) (n - 7) (n - 14) (n - 12) (n - 1) (n - 11) (n - 8)/(43046721 (3 n + 37) (3 n + 38) (3 n + 50) (3 n + 47) (3 n + 41) (3 n + 44) (3 n + 43) (3 n + 40) (n + 14) (3 n + 49) (n + 15) (n + 16) (3 n + 13) (n + 5) (3 n + 14) (n + 11) (n + 13) (3 n + 17) (3 n + 16) (n + 6) (3 n + 22) (n + 8) (3 n + 34) (3 n + 46) (3 n + 19) (n + 17) (3 n + 35) (n + 12) (3 n + 29) (3 n + 25) (n + 9) (3 n + 26) (3 n + 11) (3 n + 10) (n + 4) (3 n + 5) (3 n + 4) (n + 2) (3 n + 31) (3 n + 23) (3 n + 28) (n + 10) (n + 7) (3 n + 20) (n + 3) (3 n + 8) (3 n + 7) (3 n + 32)), 6250 (n - 13) (n - 9) (n - 16) ( 34 33 32 244140625 n + 190917968750 n + 74643945312500 n 31 30 + 19447827343750000 n + 3795140769023437500 n 29 28 + 590715588712678125000 n + 76200620497089196875000 n 27 26 + 8349768557831098323750000 n + 789714699761757100359843750 n 25 24 + 65111271074187596547731812500 n + 4704963199770709857745896210000 n 23 + 298475557300670817686093213310000 n 22 + 16600928559745826773109534636977500 n 21 + 806339106594780516257949904858485000 n 20 + 33992919733558248797332528579647777000 n 19 + 1233878196807261142616018656964638287120 n 18 + 38205643879238229696493076840954420200785 n 17 + 999233495284579781182409445153208426006254 n 16 + 21866478728478728658545291561403798476729564 n 15 + 397136883066772240002393783158541696296369632 n 14 + 5948719235874821227042049982082909543604254240 n 13 + 73150962515938950006943221689430537586881351488 n 12 + 735814882876748478024018536155583061151458553728 n 11 + 6034093881412129568061181303199971047165018868224 n 10 + 40184656718884941771650879675282816494753458504960 n 9 + 216209429325373165323638533542645311590469549747712 n 8 + 933157011089514333111211786701816648709387998538752 n 7 + 3198743737968178034888385872733837216996331630411776 n 6 + 8588738222653626476772260764753295187340301147013120 n 5 + 17715930099579986138191473897119920969270288094724096 n 4 + 27302701923524841230211676337005868780809103186067456 n 3 + 30162097806254360624178128457752657158289245756981248 n 2 + 22352659191727591595213853500879628620622263243243520 n + 9845515372852298275454272112110124347506442842931200 n + 1922984259272153980650548656445206921768725381120000) (n - 4) (n - 5) (n - 15) (n - 3) (n - 10) (n - 6) (n - 2) (n - 17) (n - 7) (n - 14) (n - 12) (n - 1) (n - 11) (n - 8)/(14348907 (3 n + 37) (3 n + 38) (3 n + 52) (3 n + 50) (3 n + 47) (n + 18) (3 n + 41) (3 n + 44) (3 n + 43) (3 n + 40) (n + 14) (3 n + 49) (n + 15) (n + 16) (3 n + 13) (n + 5) (3 n + 14) (n + 11) (n + 13) (3 n + 17) (3 n + 16) (n + 6) (3 n + 22) (n + 8) (3 n + 34) (3 n + 46) (3 n + 19) (n + 17) (3 n + 35) (n + 12) (3 n + 29) (3 n + 25) (n + 9) (3 n + 26) (3 n + 11) (3 n + 10) (n + 4) (3 n + 5) (3 n + 4) (n + 2) (3 n + 31) (3 n + 23) (3 n + 28) (n + 10) (3 n + 53) (n + 7) (3 n + 20) (n + 3) (3 n + 8) (3 n + 7) (3 n + 32))] and in Maple notation [10/3*(n-1)*(n^2+14*n+12)/(3*n+5)/(3*n+4)/(n+2), 5/3*(n-1)*(n-2)*(5*n^4+160*n^3 +1803*n^2+3768*n+2016)/(3*n+8)/(3*n+5)/(3*n+7)/(3*n+4)/(n+3)/(n+2), 20/27*(n-1) *(n-2)*(n-3)*(25*n^6+1350*n^5+31495*n^4+347406*n^3+1211092*n^2+1580304*n+665280 )/(3*n+11)/(3*n+8)/(3*n+5)/(3*n+10)/(3*n+7)/(3*n+4)/(n+4)/(n+3)/(n+2), 25/81*(n -1)*(n-2)*(n-3)*(n-4)*(125*n^8+10000*n^7+371450*n^6+7831720*n^5+90056453*n^4+ 442852216*n^3+996421764*n^2+1007115984*n+363242880)/(3*n+14)/(3*n+11)/(3*n+8)/( 3*n+5)/(3*n+13)/(3*n+10)/(3*n+7)/(3*n+4)/(n+5)/(n+4)/(n+3)/(n+2), 10/81*(n-1)*( n-2)*(n-3)*(n-4)*(n-5)*(625*n^10+68750*n^9+3628750*n^8+116901500*n^7+2395396825 *n^6+29421692462*n^5+186231365400*n^4+617582906760*n^3+1076423055120*n^2+ 918818062848*n+296406190080)/(3*n+17)/(3*n+14)/(3*n+11)/(3*n+8)/(3*n+5)/(3*n+16 )/(3*n+13)/(3*n+10)/(3*n+7)/(3*n+4)/(n+6)/(n+5)/(n+4)/(n+3)/(n+2), 175/729*(n-1 )*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*(625*n^12+90000*n^11+6325625*n^10+281268000*n^9 +8516307975*n^8+175323137472*n^7+2320161434435*n^6+17886472474464*n^5+ 79792191915340*n^4+206727295234608*n^3+302645014842240*n^2+228319317305856*n+ 67580611338240)/(3*n+20)/(3*n+17)/(3*n+14)/(3*n+11)/(3*n+8)/(3*n+5)/(3*n+19)/(3 *n+16)/(3*n+13)/(3*n+10)/(3*n+7)/(3*n+4)/(n+7)/(n+6)/(n+5)/(n+4)/(n+3)/(n+2), 200/2187*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*(n-7)*(3125*n^14+568750*n^13+ 51008125*n^12+2952031250*n^11+120616941375*n^10+3564549572010*n^9+ 75011757331015*n^8+1071261463254878*n^7+9703709965152728*n^6+54870097208217592* n^5+193681878688205040*n^4+421496433468194112*n^3+541984884627439872*n^2+ 371487057175922688*n+102587368011448320)/(3*n+23)/(3*n+20)/(3*n+17)/(3*n+14)/(3 *n+11)/(3*n+8)/(3*n+5)/(3*n+22)/(3*n+19)/(3*n+16)/(3*n+13)/(3*n+10)/(3*n+7)/(3* n+4)/(n+8)/(n+7)/(n+6)/(n+5)/(n+4)/(n+3)/(n+2), 25/729*(n-1)*(n-2)*(n-3)*(n-4)* (n-5)*(n-6)*(n-7)*(n-8)*(15625*n^16+3500000*n^15+388512500*n^14+28157710000*n^ 13+1471455483750*n^12+57589291846800*n^11+1695686514421100*n^10+ 36823499062435120*n^9+566623013737989961*n^8+5883512920147660112*n^7+ 40461585694240387992*n^6+183663017691739672800*n^5+546422776901073190224*n^4+ 1041865968637322656128*n^3+1209989486787870567168*n^2+767428785578256721920*n+ 200045367622324224000)/(3*n+26)/(3*n+23)/(3*n+20)/(3*n+17)/(3*n+14)/(3*n+11)/(3 *n+8)/(3*n+5)/(3*n+25)/(3*n+22)/(3*n+19)/(3*n+16)/(3*n+13)/(3*n+10)/(3*n+7)/(3* n+4)/(n+9)/(n+8)/(n+7)/(n+6)/(n+5)/(n+4)/(n+3)/(n+2), 250/19683*(n-1)*(n-2)*(n-\ 3)*(n-4)*(n-5)*(n-6)*(n-7)*(n-8)*(n-9)*(78125*n^18+21093750*n^17+2832000000*n^ 16+250057575000*n^15+16129151298750*n^14+795938987686500*n^13+30534485772468500 *n^12+906798973422904200*n^11+20427547655336548845*n^10+337686950884283521398*n ^9+3947955480810255108516*n^8+32043500652066554061936*n^7+ 179512767333384682604720*n^6+690539558794947366099552*n^5+ 1800054096927431745057984*n^4+3089887924189906537726464*n^3+ 3302805644601397253544960*n^2+1963602485859499681996800*n+ 487310515527981809664000)/(3*n+29)/(3*n+26)/(3*n+23)/(3*n+20)/(3*n+17)/(3*n+14) /(3*n+11)/(3*n+8)/(3*n+5)/(3*n+28)/(3*n+25)/(3*n+22)/(3*n+19)/(3*n+16)/(3*n+13) /(3*n+10)/(3*n+7)/(3*n+4)/(n+10)/(n+9)/(n+8)/(n+7)/(n+6)/(n+5)/(n+4)/(n+3)/(n+2 ), 275/59049*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*(n-7)*(n-8)*(n-9)*(n-10)*( 390625*n^20+125000000*n^19+19932421875*n^18+2100058125000*n^17+162998007281250* n^16+9812713709640000*n^15+468929848263193750*n^14+17882056172995670000*n^13+ 539717687867022031125*n^12+12647249945053650458400*n^11+ 223840869361781977857207*n^10+2906822815545027467301000*n^9+ 27223159945086515727267640*n^8+182549042700526995526230560*n^7+ 872351065103429658321105456*n^6+2945228241209805686343502080*n^5+ 6902569938133543227991752960*n^4+10868899678282135763981015040*n^3+ 10838203759365349592131264512*n^2+6098034294882387606412984320*n+ 1450236094211273865560064000)/(3*n+32)/(3*n+29)/(3*n+26)/(3*n+23)/(3*n+20)/(3*n +17)/(3*n+14)/(3*n+11)/(3*n+8)/(3*n+5)/(3*n+31)/(3*n+28)/(3*n+25)/(3*n+22)/(3*n +19)/(3*n+16)/(3*n+13)/(3*n+10)/(3*n+7)/(3*n+4)/(n+11)/(n+10)/(n+9)/(n+8)/(n+7) /(n+6)/(n+5)/(n+4)/(n+3)/(n+2), 500/59049*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*( n-7)*(n-8)*(n-9)*(n-10)*(n-11)*(390625*n^22+146093750*n^21+27263671875*n^20+ 3371893593750*n^19+308936639343750*n^18+22156447588027500*n^17+ 1279052684751943750*n^16+60115193382267897500*n^15+2299404607657354894125*n^14+ 70878598177548115189950*n^13+1729900820671159605447927*n^12+ 32666103637559910020696622*n^11+466201654576472678847544924*n^10+ 4950647702507197180824497504*n^9+38811779597851432703525662800*n^8+ 223575427080831090549810361440*n^7+940368325304867050278049974336*n^6+ 2854486677750085837425254720256*n^5+6125215851106554506656615262208*n^4+ 8968610229255342387282781237248*n^3+8427526119144033265190253772800*n^2+ 4520311863712029247503178137600*n+1035468571266849540009885696000)/(3*n+35)/(3* n+32)/(3*n+29)/(3*n+26)/(3*n+23)/(3*n+20)/(3*n+17)/(3*n+14)/(3*n+11)/(3*n+8)/(3 *n+5)/(3*n+34)/(3*n+31)/(3*n+28)/(3*n+25)/(3*n+22)/(3*n+19)/(3*n+16)/(3*n+13)/( 3*n+10)/(3*n+7)/(3*n+4)/(n+12)/(n+11)/(n+10)/(n+9)/(n+8)/(n+7)/(n+6)/(n+5)/(n+4 )/(n+3)/(n+2), 8125/531441/(3*n+37)/(3*n+38)*(390625*n^24+168750000*n^23+ 36406093750*n^22+5215711875000*n^21+555653435734375*n^20+46629094852065000*n^19 +3180008817647860000*n^18+179018258758398972000*n^17+8359337641821138259375*n^ 16+322668943357756914124800*n^15+10190093136668578130540062*n^14+ 259128925919723685281352312*n^13+5203152384672359378800158073*n^12+ 80911034292683730754499188968*n^11+960859158382204140327863395372*n^10+ 8644816687803355258802624549616*n^9+58639733337648209775385392345904*n^8+ 298365617966118297879376848036864*n^7+1129407850159607222809516005399360*n^6+ 3136439941072010200452070011439872*n^5+6245424914435425680218228384437248*n^4+ 8591178793062450532710004801671168*n^3+7666550139354642508656788490387456*n^2+ 3942762690385205796234302521344000*n+873521286720714271952339573145600)*(n-9)/( 3*n+13)/(n+5)/(3*n+14)*(n-4)/(n+11)/(n+13)/(3*n+17)/(3*n+16)/(n+6)/(3*n+22)/(n+ 8)*(n-5)/(3*n+34)*(n-3)*(n-10)*(n-6)/(3*n+19)*(n-2)/(3*n+35)/(n+12)/(3*n+29)*(n -7)/(3*n+25)/(n+9)/(3*n+26)/(3*n+11)/(3*n+10)/(n+4)*(n-12)/(3*n+5)/(3*n+4)/(n+2 )*(n-1)/(3*n+31)/(3*n+23)/(3*n+28)/(n+10)*(n-11)/(n+7)/(3*n+20)/(n+3)/(3*n+8)/( 3*n+7)/(3*n+32)*(n-8), 8750/1594323/(3*n+37)/(3*n+38)*(n-13)/(3*n+41)/(3*n+40)/ (n+14)*(n-9)/(3*n+13)/(n+5)/(3*n+14)*(n-4)/(n+11)/(n+13)/(3*n+17)/(3*n+16)/(n+6 )/(3*n+22)/(n+8)*(n-5)*(1953125*n^26+964843750*n^25+238153906250*n^24+ 39088696562500*n^23+4783301683671875*n^22+463097850821856250*n^21+ 36683217651310587500*n^20+2422106188714313530000*n^19+134451183255111956250875* n^18+6280973764252578822252250*n^17+245685651417340155574108010*n^16+ 7965254293912909544892495700*n^15+210997650380295609314709054365*n^14+ 4490896894951729029703537650022*n^13+75553740331837563548537331395264*n^12+ 992245898107799045366426545042840*n^11+10092920710035456790344883467592160*n^10 +79124024103314575984088177892139936*n^9+475943419824459295565069336112667392*n ^8+2183035723828352120366849011221763200*n^7+ 7560042126451477031820410803218136320*n^6+ 19459298282268239851120990204607649792*n^5+ 36327240522566678668419897890954563584*n^4+ 47323256974226640625530005476733583360*n^3+ 40351431834497329990516280846482145280*n^2+ 19989443234786436853023478605624115200*n+4297724730665914218005510699876352000) /(3*n+34)*(n-3)*(n-10)*(n-6)/(3*n+19)*(n-2)/(3*n+35)/(n+12)/(3*n+29)*(n-7)/(3*n +25)/(n+9)/(3*n+26)/(3*n+11)/(3*n+10)/(n+4)*(n-12)/(3*n+5)/(3*n+4)/(n+2)*(n-1)/ (3*n+31)/(3*n+23)/(3*n+28)/(n+10)*(n-11)/(n+7)/(3*n+20)/(n+3)/(3*n+8)/(3*n+7)/( 3*n+32)*(n-8), 3125/1594323/(3*n+37)/(3*n+38)*(n-13)/(3*n+41)/(3*n+44)/(3*n+43) /(3*n+40)/(n+14)/(n+15)*(n-9)/(3*n+13)/(n+5)/(3*n+14)*(n-4)/(n+11)/(n+13)/(3*n+ 17)/(3*n+16)/(n+6)/(3*n+22)/(n+8)*(n-5)/(3*n+34)*(n-3)*(n-10)*(n-6)/(3*n+19)*(n -2)/(3*n+35)*(9765625*n^28+5468750000*n^27+1530689453125*n^26+285147931250000*n ^25+39675651519765625*n^24+4381244178311000000*n^23+397761085471676265625*n^22+ 30313310875395097550000*n^21+1961174286891512081764375*n^20+ 108173359307480208653912000*n^19+5080350687698668045856094175*n^18+ 201968080938460004676743908400*n^17+6730476625682406293963951360275*n^16+ 185618071733653074203544652826240*n^15+4175519045287600533335553298262131*n^14+ 75547615625634842580331824774062672*n^13+1087185094408295711360414813396582116* n^12+12350540887794648498960848237810852432*n^11+ 110210195940320400685488505236041785488*n^10+ 769360198659614364958011391667398121856*n^9+ 4180702765066784579408128110506214560448*n^8+ 17554257240309290117180820322940103449856*n^7+ 56317168250050208409155683988784024339456*n^6+ 135716511520377820499131297787258639388672*n^5+ 239445513739029805942450641174325585575936*n^4+ 297275252835249599670681722258134702620672*n^3+ 243407207633160475975692129041365519564800*n^2+ 116590431719022121929785188974875954380800*n+ 24393885571259729101399278732498173952000)/(n+12)/(3*n+29)*(n-7)*(n-14)/(3*n+25 )/(n+9)/(3*n+26)/(3*n+11)/(3*n+10)/(n+4)*(n-12)/(3*n+5)/(3*n+4)/(n+2)*(n-1)/(3* n+31)/(3*n+23)/(3*n+28)/(n+10)*(n-11)/(n+7)/(3*n+20)/(n+3)/(3*n+8)/(3*n+7)/(3*n +32)*(n-8), 10000/14348907/(3*n+37)/(3*n+38)*(n-13)/(3*n+47)/(3*n+41)/(3*n+44)/ (3*n+43)/(3*n+40)/(n+14)/(n+15)*(n-9)/(n+16)/(3*n+13)/(n+5)/(3*n+14)*(n-4)/(n+ 11)/(n+13)/(3*n+17)/(3*n+16)/(n+6)/(3*n+22)/(n+8)*(n-5)*(n-15)/(3*n+34)*(n-3)*( n-10)/(3*n+46)*(n-6)/(3*n+19)*(n-2)*(48828125*n^30+30761718750*n^29+ 9688134765625*n^28+2031960410156250*n^27+318722301228515625*n^26+ 39764670462178593750*n^25+4093182394572664453125*n^24+355509559005491273906250* n^23+26401508061638197990734375*n^22+1687675729508032133300681250*n^21+ 93004543600428939236104858875*n^20+4406778516261193679627720358750*n^19+ 178418590210040512322228554921875*n^18+6115907790201190950611044547465250*n^17+ 175467032242094490797075924980633575*n^16+4160541182889464276961439393131583134 *n^15+80543625256002873677866111978962969600*n^14+ 1260368836589197634478427098020727091080*n^13+ 15828782284181089257133586730516581719840*n^12+ 158766249082365277105871429046512177008608*n^11+ 1266907447234675719929487284302534765679360*n^10+ 8009107483063817177214299882402164796269440*n^9+ 39883271429297735241457997296641364094027520*n^8+ 155139658233010354689023083530368301118405632*n^7+ 465613241951116716389626897049245659160289280*n^6+ 1058930175220137984397904168758941951561646080*n^5+ 1777083076464111693872158852837728810527293440*n^4+ 2113538786375237861756588628681787508248805376*n^3+ 1668577372688137461847997715434996419012853760*n^2+ 775237523777093068874655642318190772905574400*n+ 158218741815190602951675721858983156252672000)/(3*n+35)/(n+12)/(3*n+29)*(n-7)*( n-14)/(3*n+25)/(n+9)/(3*n+26)/(3*n+11)/(3*n+10)/(n+4)*(n-12)/(3*n+5)/(3*n+4)/(n +2)*(n-1)/(3*n+31)/(3*n+23)/(3*n+28)/(n+10)*(n-11)/(n+7)/(3*n+20)/(n+3)/(3*n+8) /(3*n+7)/(3*n+32)*(n-8), 53125/43046721/(3*n+37)/(3*n+38)*(48828125*n^32+ 34375000000*n^31+12098828125000*n^30+2837015937500000*n^29+497954212054687500*n ^28+69631182156532500000*n^27+8054121580992890625000*n^26+ 789081904306868932500000*n^25+66457407260279202073968750*n^24+ 4852600418265527436143700000*n^23+308331627128829839623809357000*n^22+ 17044594867995638721547301532000*n^21+816904878627420459711031509451500*n^20+ 33732480108678267613779558115932000*n^19+1189817999730359074000536545177088600* n^18+35480116109810396395927364569743253344*n^17+ 884559043914259587860093537800612890333*n^16+ 18241550264954551682697722257424720538528*n^15+ 308380288387426685676777556150201768331824*n^14+ 4245078629321314054572690168051328315823552*n^13+ 47357072559296979825413464140113357324536160*n^12+ 426547035911857122446455778738594295051708160*n^11+ 3090489064358021998406449857791159047548840192*n^10+ 17928905220873526226783376549062544302328689664*n^9+ 82744113328844656699834210313757644321233126656*n^8+ 300997423230385027747162712890582600999433220096*n^7+ 851738025635381162513359098560364285746019999744*n^6+ 1839902625040310549552774986957893370621109207040*n^5+ 2952482469677132119736370482567337249575897726976*n^4+ 3378244598858349225947722186059755141833310601216*n^3+ 2580284135802053488817923687968209123871440240640*n^2+ 1165929436813806274671864855622038676023174758400*n+ 232581550468330186338963311132705239691427840000)/(3*n+50)*(n-13)/(3*n+47)/(3*n +41)/(3*n+44)/(3*n+43)/(3*n+40)/(n+14)/(3*n+49)/(n+15)*(n-9)*(n-16)/(n+16)/(3*n +13)/(n+5)/(3*n+14)*(n-4)/(n+11)/(n+13)/(3*n+17)/(3*n+16)/(n+6)/(3*n+22)/(n+8)* (n-5)*(n-15)/(3*n+34)*(n-3)*(n-10)/(3*n+46)*(n-6)/(3*n+19)*(n-2)/(n+17)/(3*n+35 )/(n+12)/(3*n+29)*(n-7)*(n-14)/(3*n+25)/(n+9)/(3*n+26)/(3*n+11)/(3*n+10)/(n+4)* (n-12)/(3*n+5)/(3*n+4)/(n+2)*(n-1)/(3*n+31)/(3*n+23)/(3*n+28)/(n+10)*(n-11)/(n+ 7)/(3*n+20)/(n+3)/(3*n+8)/(3*n+7)/(3*n+32)*(n-8), 6250/14348907/(3*n+37)/(3*n+ 38)/(3*n+52)/(3*n+50)*(n-13)/(3*n+47)/(n+18)/(3*n+41)/(3*n+44)/(3*n+43)/(3*n+40 )/(n+14)/(3*n+49)/(n+15)*(n-9)*(n-16)/(n+16)*(244140625*n^34+190917968750*n^33+ 74643945312500*n^32+19447827343750000*n^31+3795140769023437500*n^30+ 590715588712678125000*n^29+76200620497089196875000*n^28+ 8349768557831098323750000*n^27+789714699761757100359843750*n^26+ 65111271074187596547731812500*n^25+4704963199770709857745896210000*n^24+ 298475557300670817686093213310000*n^23+16600928559745826773109534636977500*n^22 +806339106594780516257949904858485000*n^21+ 33992919733558248797332528579647777000*n^20+ 1233878196807261142616018656964638287120*n^19+ 38205643879238229696493076840954420200785*n^18+ 999233495284579781182409445153208426006254*n^17+ 21866478728478728658545291561403798476729564*n^16+ 397136883066772240002393783158541696296369632*n^15+ 5948719235874821227042049982082909543604254240*n^14+ 73150962515938950006943221689430537586881351488*n^13+ 735814882876748478024018536155583061151458553728*n^12+ 6034093881412129568061181303199971047165018868224*n^11+ 40184656718884941771650879675282816494753458504960*n^10+ 216209429325373165323638533542645311590469549747712*n^9+ 933157011089514333111211786701816648709387998538752*n^8+ 3198743737968178034888385872733837216996331630411776*n^7+ 8588738222653626476772260764753295187340301147013120*n^6+ 17715930099579986138191473897119920969270288094724096*n^5+ 27302701923524841230211676337005868780809103186067456*n^4+ 30162097806254360624178128457752657158289245756981248*n^3+ 22352659191727591595213853500879628620622263243243520*n^2+ 9845515372852298275454272112110124347506442842931200*n+ 1922984259272153980650548656445206921768725381120000)/(3*n+13)/(n+5)/(3*n+14)*( n-4)/(n+11)/(n+13)/(3*n+17)/(3*n+16)/(n+6)/(3*n+22)/(n+8)*(n-5)*(n-15)/(3*n+34) *(n-3)*(n-10)/(3*n+46)*(n-6)/(3*n+19)*(n-2)/(n+17)/(3*n+35)/(n+12)/(3*n+29)*(n-\ 17)*(n-7)*(n-14)/(3*n+25)/(n+9)/(3*n+26)/(3*n+11)/(3*n+10)/(n+4)*(n-12)/(3*n+5) /(3*n+4)/(n+2)*(n-1)/(3*n+31)/(3*n+23)/(3*n+28)/(n+10)*(n-11)/(3*n+53)/(n+7)/(3 *n+20)/(n+3)/(3*n+8)/(3*n+7)/(3*n+32)*(n-8)] and here are the, B[r], for f from 2 to , 18 10 25 500 3125 6250 109375 625000 390625 [--, ---, -----, ------, -------, ---------, -----------, -----------, 27 243 19683 531441 4782969 387420489 10460353203 31381059609 19531250 107421875 195312500 3173828125 -------------, ---------------, ----------------, ------------------, 7625597484987 205891132094649 1853020188851841 150094635296999121 17089843750 30517578125 488281250000 -------------------, --------------------, ----------------------, 4052555153018976267 36472996377170786403 2954312706550833698643 2593994140625 1525878906250 -----------------------, ------------------------] 79766443076872509863361 239299329230617529590083 and in Maple notation [10/27, 25/243, 500/19683, 3125/531441, 6250/4782969, 109375/387420489, 625000/ 10460353203, 390625/31381059609, 19531250/7625597484987, 107421875/ 205891132094649, 195312500/1853020188851841, 3173828125/150094635296999121, 17089843750/4052555153018976267, 30517578125/36472996377170786403, 488281250000 /2954312706550833698643, 2593994140625/79766443076872509863361, 1525878906250/ 239299329230617529590083] ----------------------------------------- This ends this paper that took, 1299.291, seconds to generate