What comes next?

[Gil]

Thanks for sharing this very compact code to get the coefficients list {c0 x1 c2...}.

Nice ideas yours : the while routine, the use of GDLIST (that I created) and the HEAD command (instead of 1 GET).

I don't use the NEG command in your last code,
building simply the polynomial P(x)=
c0×COMB(x 0) + c1×COMB(x 1) + c2×COMB(x 2) +... , as I understood from the Mathologer video.

My next question is if I may ask: what is the code algorithm you used in your first post to get the binomial coefficient ?
I know of course how to calculate them, but I don't understand where you calculate them in post 1, as the given code there seems to be compact. Did you use a special algorithm to have it coded in such a compact manner?

Last question, what will you get in your calculator or l'émulation for the next number of the following list?
{ 17 72 97 8 32 15 63 97 57 60 83 48 26 12 62 3 49 55 77 97 98 0 89 57 34 92 29 75 13 40 3 2 3 83 69 1 48 87 27 54 92 3 67 28 97 56 63 }

My outputs are the following on my HP50G (EMU48 on the phone):

f(x): '79452528137/73368295464161185998004072384003398572822023372800000000*x^46-62405430595537/34177777390137198446275188998759347161252495360000000000*x^45+6867032884106951/5316543149576897536087251622029231780639277056000000000*x^44-34545108562396661/63042013631346611099848833462797214790189056000000000*x^43+106691370557962873/661179349530767011079125932350358385852416000000000*x^42-923169570689950723/25870912096109496367220670975808100106240000000000*x^41+5116625404961330065763/822442605513880916220376159752884821426176000000000*x^40-311468829373514971297883/352475402363091821237304068465522066325504000000000*x^39+27594326046634577365304​3/2636033992031669603270436409464374427648000000000*x^38-87608265026221671791006869/8324317869573693484011904450940129771520000000000*x^37+3002951632494509448186313​9/32997296059571397594281423048771685580800000000*x^36-4836221201668622333612196203/70708491556224423416317335104510754816000000000*x^35+289882152645079814443270437​89639/6434472731616422530884877494510478688256000000000*x^34-839345543093035778449860512219/3205802337471230543171792686672773120000000000*x^33+5160708636241167083049878373​3509/3823216120984208277412286092994936832000000000*x^32-790243016479486246527834493165571/1274405373661402759137428697664978944000000000*x^31+1068115248911007357425117522​68222691/4193204777853647788129604101994446848000000000*x^30-8431800252037961299841608135968849877/8985438809686388117420580218559528960000000000*x^29+7682535668432642325561622458​09725666423/24725448862516336957591803497967255552000000000*x^28-1963166711585595025861899594374007832651/2119324188215686024936440299825764761600000000*x^27+2042562341192591298725383283​799475884961/82045230011366427310259378122968268800000000*x^26-63148633232678459156494019054764776365418851/104607668264492194820580707106784542720000000000*x^25+21191971223495843964537533​74358695387711789651/160398424672221365391557084230402965504000000000*x^24-17858444764609635577090499051547310644173/68423124970642479286895385772032000000000*x^23+575125831754572115635515365719990​27424387232007/12362724431258168478795901748983627776000000000*x^22-43054411737196706236305010020807767010248926001/575903312015132071993597907561349120000000000*x^21+11146376398461744022298118158​88516792729194246171/1030227035938180706566325145748635648000000000*x^20-572545195255310504916372668049778639167342175967/40666856681770291048670729437446144000000000*x^19+161570098669313148872847271431​80289175362272256437/98278236980944870034287596140494848000000000*x^18-1252209966295262840527254340195582349765162487367/728706650872057859868667297136640000000000*x^17+47399261150573426840777142764422​79198720541863850271/295701871857372358956106502505077145600000000*x^16-168492124329437939303112790767488353520865947410307863/1267293736531595824097599296450330624000000000*x^15+2097376832799371152402216895​03251905819425068750689473/214826146221167953087769681307107328000000000*x^14-924867262101770565596196284887085917299220361455178393/146472372423523604378024782709391360000000000*x^13+21545644845605013092131446491​07615993811498231839781/60276696470585845423055466135552000000000*x^12-111065175127946654779403925649034914832194926918919/631782144683935491623640367104000000000*x^11+30151778145877484822280521461025808​81633138423086659489/4050505360622676774421352242741248000000000*x^10-14144171311503183171149392051950453414030477549299293597/5269790137544809068762473581117440000000000*x^9+12145489715007185763814880131700​4578091817898013556907/14943270644658312752856551242752000000000*x^8-728465154287934961017826027803007940038213284428022953/35863849547179950606855722982604800000000*x^7+8175783823440875967323115918963477​00161404311307203053/19968637217010071263817198845870080000000*x^6-4359514147282944414325206668958078534305004412820119/67563058253041594501637138952192000000*x^5+9251853218738204433802104415183867205​3095671664831/1216909580968584274864143353702400000*x^4-49357091850914086397470724236043481620906093159/790201026602976801859833346560000*x^3+132720057639127275173051837261789837516351​/4202885913130495995387984000*x^2-6869689147511982724488320657561/941958815880242160*x+17'

f(47): -313323742215540

Thanks in advance for your collaboration, Thomas.

Regards,
Gil