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, 11 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) (-5 + n) (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)*(-5+n)*(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) (-5 + n) (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)*(-5+n)*(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) (-5 + n) (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)*(-5+n)*(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) (-5 + n) (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)*(-5+n)*(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) (-5 + n) (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)*(-5+n)*(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) (-5 + n) (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)*(-5+n)*(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 To sum up, here are the, A[r, n], for f from 2 to , 11 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 10 (n - 1) (n - 2) (n - 3) (n - 4) (-5 + n) (625 n + 68750 n 8 7 6 5 + 3628750 n + 116901500 n + 2395396825 n + 29421692462 n 4 3 2 + 186231365400 n + 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)), 175 (n - 1) (n - 2) (n - 3) (n - 4) (-5 + n) 12 11 10 9 8 (n - 6) (625 n + 90000 n + 6325625 n + 281268000 n + 8516307975 n 7 6 5 + 175323137472 n + 2320161434435 n + 17886472474464 n 4 3 2 + 79792191915340 n + 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) (n - 2) (n - 3) (n - 4) (-5 + n) (n - 6) (n - 7) ( 14 13 12 11 10 3125 n + 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)), 25 (n - 1) (n - 2) (n - 3) (n - 4) (-5 + n) (n - 6) 16 15 14 13 (n - 7) (n - 8) (15625 n + 3500000 n + 388512500 n + 28157710000 n 12 11 10 + 1471455483750 n + 57589291846800 n + 1695686514421100 n 9 8 7 + 36823499062435120 n + 566623013737989961 n + 5883512920147660112 n 6 5 + 40461585694240387992 n + 183663017691739672800 n 4 3 + 546422776901073190224 n + 1041865968637322656128 n 2 + 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) 18 (n - 3) (n - 4) (-5 + n) (n - 6) (n - 7) (n - 8) (n - 9) (78125 n 17 16 15 14 + 21093750 n + 2832000000 n + 250057575000 n + 16129151298750 n 13 12 11 + 795938987686500 n + 30534485772468500 n + 906798973422904200 n 10 9 + 20427547655336548845 n + 337686950884283521398 n 8 7 + 3947955480810255108516 n + 32043500652066554061936 n 6 5 + 179512767333384682604720 n + 690539558794947366099552 n 4 3 + 1800054096927431745057984 n + 3089887924189906537726464 n 2 + 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) (-5 + n) (n - 6) (n - 7) 20 19 18 (n - 8) (n - 9) (n - 10) (390625 n + 125000000 n + 19932421875 n 17 16 15 + 2100058125000 n + 162998007281250 n + 9812713709640000 n 14 13 + 468929848263193750 n + 17882056172995670000 n 12 11 + 539717687867022031125 n + 12647249945053650458400 n 10 9 + 223840869361781977857207 n + 2906822815545027467301000 n 8 7 + 27223159945086515727267640 n + 182549042700526995526230560 n 6 5 + 872351065103429658321105456 n + 2945228241209805686343502080 n 4 3 + 6902569938133543227991752960 n + 10868899678282135763981015040 n 2 + 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))] 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)*(-5+n)*(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)*(-5+n)*(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)*(-5+n)*(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)* (-5+n)*(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)*(-5+n)*(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)*(-5+n)*(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)] and here are the, B[r], for f from 2 to , 11 10 25 500 3125 6250 109375 625000 390625 [--, ---, -----, ------, -------, ---------, -----------, -----------, 27 243 19683 531441 4782969 387420489 10460353203 31381059609 19531250 107421875 -------------, ---------------] 7625597484987 205891132094649 and in Maple notation [10/27, 25/243, 500/19683, 3125/531441, 6250/4782969, 109375/387420489, 625000/ 10460353203, 390625/31381059609, 19531250/7625597484987, 107421875/ 205891132094649] ----------------------------------------- This ends this paper that took, 24.618, seconds to generate