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Where each entry of Gegenbauer polynomials came from

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339 entries, page 1 of 2. next

alphan value last changed by
a 0 1 2026-08-09 — flattening
a 1 2*a*x 2026-08-09 — flattening
a 2 2*(a + 1)*a*x^2 - a 2026-08-09 — flattening
a 3 4/3*(a + 2)*(a + 1)*a*x^3 - 2*(a + 1)*a*x 2026-08-09 — flattening
a 4 2/3*(a + 3)*(a + 2)*(a + 1)*a*x^4 - 2*(a + 2)*(a + 1)*a*x^2 + 1/2*(a … 2026-08-09 — flattening
a 5 4/15*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^5 - 4/3*(a + 3)*(a + 2)*(a +… 2026-08-09 — flattening
a 6 4/45*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^6 - 2/3*(a + 4)*(a +… 2026-08-09 — flattening
a 7 8/315*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^7 - 4/15*(a… 2026-08-09 — flattening
a 8 2/315*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1)*a*x^8 -… 2026-08-09 — flattening
a 9 4/2835*(a + 8)*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + 2)*(a + 1… 2026-08-09 — flattening
a 10 4/14175*(a + 9)*(a + 8)*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a + 3)*(a + … 2026-08-09 — flattening
a 11 8/155925*(a + 10)*(a + 9)*(a + 8)*(a + 7)*(a + 6)*(a + 5)*(a + 4)*(a … 2026-08-09 — flattening
a 12 4/467775*(a + 11)*(a + 10)*(a + 9)*(a + 8)*(a + 7)*(a + 6)*(a + 5)*(a… 2026-08-09 — flattening
a 13 8/6081075*(a + 12)*(a + 11)*(a + 10)*(a + 9)*(a + 8)*(a + 7)*(a + 6)*… 2026-08-09 — flattening
a 14 8/42567525*(a + 13)*(a + 12)*(a + 11)*(a + 10)*(a + 9)*(a + 8)*(a + 7… 2026-08-09 — flattening
a 15 16/638512875*(a + 14)*(a + 13)*(a + 12)*(a + 11)*(a + 10)*(a + 9)*(a … 2026-08-09 — flattening
-1 0 1 2026-08-09 — flattening
-1 1 -2*x 2026-08-09 — flattening
-1 2 1 2026-08-09 — flattening
-1 3 0 2026-08-09 — flattening
-3/4 0 1 2026-08-09 — flattening
-3/4 1 -3/2*x 2026-08-09 — flattening
-3/4 2 -3/8*x^2 + 3/4 2026-08-09 — flattening
-3/4 3 -5/16*x^3 + 3/8*x 2026-08-09 — flattening
-3/4 4 -45/128*x^4 + 15/32*x^2 - 3/32 2026-08-09 — flattening
-3/4 5 -117/256*x^5 + 45/64*x^3 - 15/64*x 2026-08-09 — flattening
-3/4 6 -663/1024*x^6 + 585/512*x^4 - 135/256*x^2 + 5/128 2026-08-09 — flattening
-3/4 7 -1989/2048*x^7 + 1989/1024*x^5 - 585/512*x^3 + 45/256*x 2026-08-09 — flattening
-3/4 8 -49725/32768*x^8 + 13923/4096*x^6 - 9945/4096*x^4 + 585/1024*x^2 - 45… 2026-08-09 — flattening
-3/4 9 -160225/65536*x^9 + 49725/8192*x^7 - 41769/8192*x^5 + 3315/2048*x^3 -… 2026-08-09 — flattening
-3/4 10 -1057485/262144*x^10 + 1442025/131072*x^8 - 348075/32768*x^6 + 69615/… 2026-08-09 — flattening
-3/4 11 -3556995/524288*x^11 + 5287425/262144*x^9 - 1442025/65536*x^7 + 34807… 2026-08-09 — flattening
-3/4 12 -48612265/4194304*x^12 + 39126945/1048576*x^10 - 47586825/1048576*x^8… 2026-08-09 — flattening
-3/4 13 -168273225/8388608*x^13 + 145836795/2097152*x^11 - 195634725/2097152*… 2026-08-09 — flattening
-3/4 14 -1177912575/33554432*x^14 + 2187551925/16777216*x^12 - 1604204745/838… 2026-08-09 — flattening
-3/4 15 -4161957765/67108864*x^15 + 8245388025/33554432*x^13 - 6562655775/167… 2026-08-09 — flattening
-3/4 16 -237231592605/2147483648*x^16 + 62429366475/134217728*x^14 - 10719004… 2026-08-09 — flattening
-3/4 17 -851242773465/4294967296*x^17 + 237231592605/268435456*x^15 - 4370055… 2026-08-09 — flattening
-3/4 18 -6147864475025/17179869184*x^18 + 14471127148905/8589934592*x^16 - 35… 2026-08-09 — flattening
-3/4 19 -22326455198775/34359738368*x^19 + 55330780275225/17179869184*x^17 - … 2026-08-09 — flattening
-3/4 20 -325966245902115/274877906944*x^20 + 424202648776725/68719476736*x^18… 2026-08-09 — flattening
-2/3 0 1 2026-08-09 — flattening
-2/3 1 -4/3*x 2026-08-09 — flattening
-2/3 2 -4/9*x^2 + 2/3 2026-08-09 — flattening
-2/3 3 -32/81*x^3 + 4/9*x 2026-08-09 — flattening
-2/3 4 -112/243*x^4 + 16/27*x^2 - 1/9 2026-08-09 — flattening
-2/3 5 -448/729*x^5 + 224/243*x^3 - 8/27*x 2026-08-09 — flattening
-2/3 6 -5824/6561*x^6 + 1120/729*x^4 - 56/81*x^2 + 4/81 2026-08-09 — flattening
-2/3 7 -26624/19683*x^7 + 5824/2187*x^5 - 1120/729*x^3 + 56/243*x 2026-08-09 — flattening
-2/3 8 -126464/59049*x^8 + 93184/19683*x^6 - 7280/2187*x^4 + 560/729*x^2 - 7… 2026-08-09 — flattening
-2/3 9 -5564416/1594323*x^9 + 505856/59049*x^7 - 46592/6561*x^5 + 14560/6561… 2026-08-09 — flattening
-2/3 10 -27822080/4782969*x^10 + 2782208/177147*x^8 - 885248/59049*x^6 + 1164… 2026-08-09 — flattening
-2/3 11 -141639680/14348907*x^11 + 139110400/4782969*x^9 - 5564416/177147*x^7… 2026-08-09 — flattening
-2/3 12 -2195415040/129140163*x^12 + 779018240/14348907*x^10 - 34777600/53144… 2026-08-09 — flattening
-2/3 13 -11483709440/387420489*x^13 + 4390830080/43046721*x^11 - 1947545600/1… 2026-08-09 — flattening
-2/3 14 -60699607040/1162261467*x^14 + 74644111360/387420489*x^12 - 120747827… 2026-08-09 — flattening
-2/3 15 -971193712640/10460353203*x^15 + 424897249280/1162261467*x^13 - 74644… 2026-08-09 — flattening
-2/3 16 -5220166205440/31381059609*x^16 + 2427984281600/3486784401*x^14 - 138… 2026-08-09 — flattening
-2/3 17 -28250311229440/94143178827*x^17 + 41761329643520/31381059609*x^15 - … 2026-08-09 — flattening
-2/3 18 -1384265250242560/2541865828329*x^18 + 240127645450240/94143178827*x^… 2026-08-09 — flattening
-2/3 19 -7577030843432960/7625597484987*x^19 + 1384265250242560/282429536481*… 2026-08-09 — flattening
-2/3 20 -41673669638881280/22876792454961*x^20 + 71981793012613120/7625597484… 2026-08-09 — flattening
-1/2 0 1 2026-08-09 — flattening
-1/2 1 -x 2026-08-09 — flattening
-1/2 2 -1/2*x^2 + 1/2 2026-08-09 — flattening
-1/2 3 -1/2*x^3 + 1/2*x 2026-08-09 — flattening
-1/2 4 -5/8*x^4 + 3/4*x^2 - 1/8 2026-08-09 — flattening
-1/2 5 -7/8*x^5 + 5/4*x^3 - 3/8*x 2026-08-09 — flattening
-1/2 6 -21/16*x^6 + 35/16*x^4 - 15/16*x^2 + 1/16 2026-08-09 — flattening
-1/2 7 -33/16*x^7 + 63/16*x^5 - 35/16*x^3 + 5/16*x 2026-08-09 — flattening
-1/2 8 -429/128*x^8 + 231/32*x^6 - 315/64*x^4 + 35/32*x^2 - 5/128 2026-08-09 — flattening
-1/2 9 -715/128*x^9 + 429/32*x^7 - 693/64*x^5 + 105/32*x^3 - 35/128*x 2026-08-09 — flattening
-1/2 10 -2431/256*x^10 + 6435/256*x^8 - 3003/128*x^6 + 1155/128*x^4 - 315/256… 2026-08-09 — flattening
-1/2 11 -4199/256*x^11 + 12155/256*x^9 - 6435/128*x^7 + 3003/128*x^5 - 1155/2… 2026-08-09 — flattening
-1/2 12 -29393/1024*x^12 + 46189/512*x^10 - 109395/1024*x^8 + 15015/256*x^6 -… 2026-08-09 — flattening
-1/2 13 -52003/1024*x^13 + 88179/512*x^11 - 230945/1024*x^9 + 36465/256*x^7 -… 2026-08-09 — flattening
-1/2 14 -185725/2048*x^14 + 676039/2048*x^12 - 969969/2048*x^10 + 692835/2048… 2026-08-09 — flattening
-1/2 15 -334305/2048*x^15 + 1300075/2048*x^13 - 2028117/2048*x^11 + 1616615/2… 2026-08-09 — flattening
-1/2 16 -9694845/32768*x^16 + 5014575/4096*x^14 - 16900975/8192*x^12 + 743642… 2026-08-09 — flattening
-1/2 17 -17678835/32768*x^17 + 9694845/4096*x^15 - 35102025/8192*x^13 + 16900… 2026-08-09 — flattening
-1/2 18 -64822395/65536*x^18 + 300540195/65536*x^16 - 145422675/16384*x^14 + … 2026-08-09 — flattening
-1/2 19 -119409675/65536*x^19 + 583401555/65536*x^17 - 300540195/16384*x^15 +… 2026-08-09 — flattening
-1/2 20 -883631595/262144*x^20 + 2268783825/131072*x^18 - 9917826435/262144*x… 2026-08-09 — flattening
-1/3 0 1 2026-08-09 — flattening
-1/3 1 -2/3*x 2026-08-09 — flattening
-1/3 2 -4/9*x^2 + 1/3 2026-08-09 — flattening
-1/3 3 -40/81*x^3 + 4/9*x 2026-08-09 — flattening
-1/3 4 -160/243*x^4 + 20/27*x^2 - 1/9 2026-08-09 — flattening
-1/3 5 -704/729*x^5 + 320/243*x^3 - 10/27*x 2026-08-09 — flattening
-1/3 6 -9856/6561*x^6 + 1760/729*x^4 - 80/81*x^2 + 5/81 2026-08-09 — flattening
-1/3 7 -47872/19683*x^7 + 9856/2187*x^5 - 1760/729*x^3 + 80/243*x 2026-08-09 — flattening
-1/3 8 -239360/59049*x^8 + 167552/19683*x^6 - 12320/2187*x^4 + 880/729*x^2 -… 2026-08-09 — flattening
-1/3 9 -11010560/1594323*x^9 + 957440/59049*x^7 - 83776/6561*x^5 + 24640/656… 2026-08-09 — flattening
-1/3 10 -57254912/4782969*x^10 + 5505280/177147*x^8 - 1675520/59049*x^6 + 209… 2026-08-09 — flattening
-1/3 11 -301889536/14348907*x^11 + 286274560/4782969*x^9 - 11010560/177147*x^… 2026-08-09 — flattening
-1/3 12 -4830232576/129140163*x^12 + 1660392448/14348907*x^10 - 71568640/5314… 2026-08-09 — flattening
-1/3 13 -26008944640/387420489*x^13 + 9660465152/43046721*x^11 - 4150981120/1… 2026-08-09 — flattening
-1/3 14 -141191413760/1162261467*x^14 + 169058140160/387420489*x^12 - 2656627… 2026-08-09 — flattening
-1/3 15 -2315539185664/10460353203*x^15 + 988339896320/1162261467*x^13 - 1690… 2026-08-09 — flattening
-1/3 16 -12735465521152/31381059609*x^16 + 5788847964160/3486784401*x^14 - 32… 2026-08-09 — flattening
-1/3 17 -70419632881664/94143178827*x^17 + 101883724169216/31381059609*x^15 -… 2026-08-09 — flattening
-1/3 18 -3520981644083200/2541865828329*x^18 + 598566879494144/94143178827*x^… 2026-08-09 — flattening
-1/3 19 -19643371277516800/7625597484987*x^19 + 3520981644083200/282429536481… 2026-08-09 — flattening
-1/3 20 -110002879154094080/22876792454961*x^20 + 186612027136409600/76255974… 2026-08-09 — flattening
-1/4 0 1 2026-08-09 — flattening
-1/4 1 -1/2*x 2026-08-09 — flattening
-1/4 2 -3/8*x^2 + 1/4 2026-08-09 — flattening
-1/4 3 -7/16*x^3 + 3/8*x 2026-08-09 — flattening
-1/4 4 -77/128*x^4 + 21/32*x^2 - 3/32 2026-08-09 — flattening
-1/4 5 -231/256*x^5 + 77/64*x^3 - 21/64*x 2026-08-09 — flattening
-1/4 6 -1463/1024*x^6 + 1155/512*x^4 - 231/256*x^2 + 7/128 2026-08-09 — flattening
-1/4 7 -4807/2048*x^7 + 4389/1024*x^5 - 1155/512*x^3 + 77/256*x 2026-08-09 — flattening
-1/4 8 -129789/32768*x^8 + 33649/4096*x^6 - 21945/4096*x^4 + 1155/1024*x^2 -… 2026-08-09 — flattening
-1/4 9 -447051/65536*x^9 + 129789/8192*x^7 - 100947/8192*x^5 + 7315/2048*x^3… 2026-08-09 — flattening
-1/4 10 -3129357/262144*x^10 + 4023459/131072*x^8 - 908523/32768*x^6 + 168245… 2026-08-09 — flattening
-1/4 11 -11094993/524288*x^11 + 15646785/262144*x^9 - 4023459/65536*x^7 + 908… 2026-08-09 — flattening
-1/4 12 -159028233/4194304*x^12 + 122044923/1048576*x^10 - 140821065/1048576*… 2026-08-09 — flattening
-1/4 13 -574948227/8388608*x^13 + 477084699/2097152*x^11 - 610224615/2097152*… 2026-08-09 — flattening
-1/4 14 -4188908511/33554432*x^14 + 7474326951/16777216*x^12 - 5247931689/838… 2026-08-09 — flattening
-1/4 15 -15359331207/67108864*x^15 + 29322359577/33554432*x^13 - 22422980853/… 2026-08-09 — flattening
-1/4 16 -906200541213/2147483648*x^16 + 230389968105/134217728*x^14 - 3811906… 2026-08-09 — flattening
-1/4 17 -3358272593907/4294967296*x^17 + 906200541213/268435456*x^15 - 161272… 2026-08-09 — flattening
-1/4 18 -25000473754641/17179869184*x^18 + 57090634096419/8589934592*x^16 - 1… 2026-08-09 — flattening
-1/4 19 -93422822977869/34359738368*x^19 + 225004263791769/17179869184*x^17 -… 2026-08-09 — flattening
-1/4 20 -1401342344668035/274877906944*x^20 + 1775033636579511/68719476736*x^… 2026-08-09 — flattening
0 0 1 2026-08-09 — flattening
0 1 0 2026-08-09 — flattening
1/4 0 1 2026-08-09 — flattening
1/4 1 1/2*x 2026-08-09 — flattening
1/4 2 5/8*x^2 - 1/4 2026-08-09 — flattening
1/4 3 15/16*x^3 - 5/8*x 2026-08-09 — flattening
1/4 4 195/128*x^4 - 45/32*x^2 + 5/32 2026-08-09 — flattening
1/4 5 663/256*x^5 - 195/64*x^3 + 45/64*x 2026-08-09 — flattening
1/4 6 4641/1024*x^6 - 3315/512*x^4 + 585/256*x^2 - 15/128 2026-08-09 — flattening
1/4 7 16575/2048*x^7 - 13923/1024*x^5 + 3315/512*x^3 - 195/256*x 2026-08-09 — flattening
1/4 8 480675/32768*x^8 - 116025/4096*x^6 + 69615/4096*x^4 - 3315/1024*x^2 +… 2026-08-09 — flattening
1/4 9 1762475/65536*x^9 - 480675/8192*x^7 + 348075/8192*x^5 - 23205/2048*x^… 2026-08-09 — flattening
1/4 10 13042315/262144*x^10 - 15862275/131072*x^8 + 3364725/32768*x^6 - 5801… 2026-08-09 — flattening
1/4 11 48612265/524288*x^11 - 65211575/262144*x^9 + 15862275/65536*x^7 - 336… 2026-08-09 — flattening
1/4 12 729183975/4194304*x^12 - 534734915/1048576*x^10 + 586904175/1048576*x… 2026-08-09 — flattening
1/4 13 2748462675/8388608*x^13 - 2187551925/2097152*x^11 + 2673674575/209715… 2026-08-09 — flattening
1/4 14 20809788825/33554432*x^14 - 35730014775/16777216*x^12 + 24063071175/8… 2026-08-09 — flattening
1/4 15 79077197535/67108864*x^15 - 145668521775/33554432*x^13 + 107190044325… 2026-08-09 — flattening
1/4 16 4823709049635/2147483648*x^16 - 1186157963025/134217728*x^14 + 189369… 2026-08-09 — flattening
1/4 17 18443593425075/4294967296*x^17 - 4823709049635/268435456*x^15 + 83031… 2026-08-09 — flattening
1/4 18 141400882925575/17179869184*x^18 - 313541088226275/8589934592*x^16 + … 2026-08-09 — flattening
1/4 19 543277076503525/34359738368*x^19 - 1272607946330175/17179869184*x^17 … 2026-08-09 — flattening
1/4 20 8366466978154285/274877906944*x^20 - 10322264453566975/68719476736*x^… 2026-08-09 — flattening
1/3 0 1 2026-08-09 — flattening
1/3 1 2/3*x 2026-08-09 — flattening
1/3 2 8/9*x^2 - 1/3 2026-08-09 — flattening
1/3 3 112/81*x^3 - 8/9*x 2026-08-09 — flattening
1/3 4 560/243*x^4 - 56/27*x^2 + 2/9 2026-08-09 — flattening
1/3 5 2912/729*x^5 - 1120/243*x^3 + 28/27*x 2026-08-09 — flattening
1/3 6 46592/6561*x^6 - 7280/729*x^4 + 280/81*x^2 - 14/81 2026-08-09 — flattening
1/3 7 252928/19683*x^7 - 46592/2187*x^5 + 7280/729*x^3 - 280/243*x 2026-08-09 — flattening
1/3 8 1391104/59049*x^8 - 885248/19683*x^6 + 58240/2187*x^4 - 3640/729*x^2 … 2026-08-09 — flattening
1/3 9 69555200/1594323*x^9 - 5564416/59049*x^7 + 442624/6561*x^5 - 116480/6… 2026-08-09 — flattening
1/3 10 389509120/4782969*x^10 - 34777600/177147*x^8 + 9737728/59049*x^6 - 11… 2026-08-09 — flattening
1/3 11 2195415040/14348907*x^11 - 1947545600/4782969*x^9 + 69555200/177147*x… 2026-08-09 — flattening
1/3 12 37322055680/129140163*x^12 - 12074782720/14348907*x^10 + 486886400/53… 2026-08-09 — flattening
1/3 13 212448624640/387420489*x^13 - 74644111360/43046721*x^11 + 30186956800… 2026-08-09 — flattening
1/3 14 1213992140800/1162261467*x^14 - 1380916060160/387420489*x^12 + 205271… 2026-08-09 — flattening
1/3 15 20880664821760/10460353203*x^15 - 8497944985600/1162261467*x^13 + 138… 2026-08-09 — flattening
1/3 16 120063822725120/31381059609*x^16 - 52201662054400/3486784401*x^14 + 2… 2026-08-09 — flattening
1/3 17 692132625121280/94143178827*x^17 - 960510581800960/31381059609*x^15 +… 2026-08-09 — flattening
1/3 18 35990896506306560/2541865828329*x^18 - 5883127313530880/94143178827*x… 2026-08-09 — flattening
1/3 19 208368348194406400/7625597484987*x^19 - 35990896506306560/28242953648… 2026-08-09 — flattening
1/3 20 1208536419527557120/22876792454961*x^20 - 1979499307846860800/7625597… 2026-08-09 — flattening
1/2 None —
2/3 0 1 2026-08-09 — flattening
2/3 1 4/3*x 2026-08-09 — flattening
2/3 2 20/9*x^2 - 2/3 2026-08-09 — flattening
2/3 3 320/81*x^3 - 20/9*x 2026-08-09 — flattening
2/3 4 1760/243*x^4 - 160/27*x^2 + 5/9 2026-08-09 — flattening
2/3 5 9856/729*x^5 - 3520/243*x^3 + 80/27*x 2026-08-09 — flattening
2/3 6 167552/6561*x^6 - 24640/729*x^4 + 880/81*x^2 - 40/81 2026-08-09 — flattening
2/3 7 957440/19683*x^7 - 167552/2187*x^5 + 24640/729*x^3 - 880/243*x 2026-08-09 — flattening
2/3 8 5505280/59049*x^8 - 3351040/19683*x^6 + 209440/2187*x^4 - 12320/729*x… 2026-08-09 — flattening
2/3 9 286274560/1594323*x^9 - 22021120/59049*x^7 + 1675520/6561*x^5 - 41888… 2026-08-09 — flattening
2/3 10 1660392448/4782969*x^10 - 143137280/177147*x^8 + 38536960/59049*x^6 -… 2026-08-09 — flattening
2/3 11 9660465152/14348907*x^11 - 8301962240/4782969*x^9 + 286274560/177147*… 2026-08-09 — flattening
2/3 12 169058140160/129140163*x^12 - 53132558336/14348907*x^10 + 2075490560/… 2026-08-09 — flattening
2/3 13 988339896320/387420489*x^13 - 338116280320/43046721*x^11 + 1328313958… 2026-08-09 — flattening
2/3 14 5788847964160/1162261467*x^14 - 6424209326080/387420489*x^12 + 929819… 2026-08-09 — flattening
2/3 15 101883724169216/10460353203*x^15 - 40521935749120/1162261467*x^13 + 6… 2026-08-09 — flattening
2/3 16 598566879494144/31381059609*x^16 - 254709310423040/3486784401*x^14 + … 2026-08-09 — flattening
2/3 17 3520981644083200/94143178827*x^17 - 4788535035953152/31381059609*x^15… 2026-08-09 — flattening
2/3 18 186612027136409600/2541865828329*x^18 - 29928343974707200/94143178827… 2026-08-09 — flattening
2/3 19 1100028791540940800/7625597484987*x^19 - 186612027136409600/282429536… 2026-08-09 — flattening
2/3 20 6490169870091550720/22876792454961*x^20 - 10450273519638937600/762559… 2026-08-09 — flattening
3/4 0 1 2026-08-09 — flattening
3/4 1 3/2*x 2026-08-09 — flattening
3/4 2 21/8*x^2 - 3/4 2026-08-09 — flattening
3/4 3 77/16*x^3 - 21/8*x 2026-08-09 — flattening
3/4 4 1155/128*x^4 - 231/32*x^2 + 21/32 2026-08-09 — flattening
3/4 5 4389/256*x^5 - 1155/64*x^3 + 231/64*x 2026-08-09 — flattening
3/4 6 33649/1024*x^6 - 21945/512*x^4 + 3465/256*x^2 - 77/128 2026-08-09 — flattening
3/4 7 129789/2048*x^7 - 100947/1024*x^5 + 21945/512*x^3 - 1155/256*x 2026-08-09 — flattening
3/4 8 4023459/32768*x^8 - 908523/4096*x^6 + 504735/4096*x^4 - 21945/1024*x^… 2026-08-09 — flattening