######################################################################
## AperyLimits Save this file as AperyLimits.txt  to use it,         #
# stay in the                                                        #
## same directory, get into Maple (by typing: maple <Enter> )        #
## and then type:  read `AperyLimits.txt` <Enter>                    #
## Then follow the instructions given there                          #
##                                                                   #
## Written by Doron Zeilberger, Rutgers University ,                 #
##  DoronZeil at gmail dot com                                       # 
######################################################################


print(`First Written: April 2021: tested for Maple 2018 `):
print(`Version  : April 2021:  `):
print():
print(`This is AperyLimit.txt, A Maple package`):
print(`accompanying Robert Dougherty-Bliss  and Doron Zeilberger's  article: `):
print(`" Experimenting with Apery Limits and WZ pairs" `):

print():
print(`The most current version is available on WWW at:`):
print(` http://sites.math.rutgers.edu/~zeilberg/tokhniot/AperyLimits.txt .`):
print(`Please report all bugs to: DoronZeil at gmail dot com .`):

print(`---------------------------`):
print():
print(`For general help, and a list of the MAIN functions,`):
print(` type "ezra();". For specific help type "ezra(procedure_name);" `):

print(`---------------------------`):

print(`For a list of the supporting functions type: ezra1();`):
print(` For specific help type "ezra(procedure_name);" `):
print():

print(`---------------------------`):


print(`---------------------------`):

print(`For a list of the STORY functions type: ezraS();`):
print(` For specific help type "ezra(procedure_name);" `):
print():

print(`---------------------------`):


print(`For a list of the Pre-computed functions type: ezraPC();`):
print(` For specific help type "ezra(procedure_name);" `):
print():

print(`---------------------------`):
 
print(`For help with EKHAD procedures type: ezraEKHAD();`):
print():

ezra1:=proc()
if args=NULL then

print(`The SUPPORTING procedures are`):
print(` MyIDs, MyIDsLog, OperAZd, OperZeil, SeqFromRec, GEk, guessGEk, guessPol`):

else
ezra(args):

fi:

end:


ezraS:=proc()
if args=NULL then

print(`The STORY procedures are`):
print(`CubicStory, DELazSipur, DELazV,  DELazGv, DELsipur1, DELsipur2, DELsipur3, DELv, QuadStory `):

else
ezra(args):

fi:

end:



ezraPC:=proc()
if args=NULL then

print(`The Pre-computed procedures are`):
print(` OutAZ1, OutF1 `):

else
ezra(args):

fi:

end:


ezra:=proc()
if args=NULL then

print(`The MAIN procedures are:`):
print(` DEL, DELaz, DELazG,  DELmaz, FindAZ1, FindAZ2, FindMAZ, FindZ1, FindZ2, OpeMAZ1, RAT, RATaz `):


elif nargs=1 and args[1]=CubicStory then
print(`CubicStory(c): An article about conjectured fast approximations to the root of 64+144*x*(1+2*c)+108*x^2*(3*c^2+3*c+1)+27*x^3*(1+2*c)=0. Try:`):
print(`CubicStory(c):`):

elif nargs=1 and args[1]=DEL then
print(`DEL(F,k,n,K): inputs a closed-form expression and outputs  the sequence a(n)/b(n) where`):
print(`b(n):=Sum(F,k=0..n) and a(n) is satisfies the same initial conditions  with a(i)=0  for negative i and 0 and a(1)=1, `):
print(`It also returns the deltas. Try:`):
print(`DEL(binomial(n,k)*binomial(n+k,k)*binomial(n+2*k,k),k,n,1000);`):
print(`DEL(binomial(n,k)*binomial(n+k,k)*binomial(n+2*k,k)*a^k,k,n,1000);, for a=2,3,4,`):
print(`DEL(binomial(n,k)*binomial(n+2*k,k)*a^k,k,n,1000);, for a=3,4,`):
print(` DEL(binomial(n,k)*binomial(2*n+2*k,2*k)*3^k,k,n,1000);`):
print(`DEL(binomial(n-k,2*k)*a^(n-k),k,n,1000);, a=1,2,3,...`):
print(`DEL(binomial(n,k)*binomial(n+k,k)^2*a^k,k,n,1000);, a=1,2,3,...`):
print(` DEL(binomial(n,k)^2*binomial(n+2*k,2*k)*a^k,k,n,1000);, a=1,2,3,4,...`):
print(` DEL(binomial(n,k)^3*binomial(n+k,k)^2*a^k,k,n,1000);, a=1,2,3...`):



elif nargs=1 and args[1]=DELaz then
print(`DELaz(P,x,K): inputs a Laurent polynoimal P in the cont. variable x and the discrete variable n`):
print(`and outputs  a(K)/b(K) where`):
print(`b(n):=CT(P^n,x) and a(n) is satisfies the same initial conditions  with a(i)=0  for negative i and 0 and a(1)=1`):
print(`It also gives you the estimated delta. try:`):
print(`DELaz(1/x+1+x,x,1000);`):
print(`a:=1;DELaz((x+1)*((a+1)*x+a)/x,x,1000);identify(%[1]); and for larger a, getting (conjecturally) arctanh(1/(2*a+1))`):


elif nargs=1 and args[1]=DELazSipur then
print(`DELazSipur(x,K): outputs an article with the DELazV(P,x,K) for P in OutAZ1(x), giving positive deltas. Try:`):
print(`DELazSipur(x,1000):`):

elif nargs=1 and args[1]=DELazV then
print(`DELazV(P,x,K): a verbose form of DELaz(P,x,K) (q.v.), try:`):
print(`DELazV(1/x+1+x,x,1000);`):
print(`a:=1;DELazV((x+1)*((a+1)*x+a)/x,x,1000);identify(%[1]); and for larger a, getting (conjecturally) arctanh(1/(2*a+1))`):

elif nargs=1 and args[1]=DELsipur1 then
print(`DELsipur1(A,B,k,n,K): outputs an article for DEL(F,k,n,K) for all F of the form binomial(n,k)*binomial(a1*n+b1*k,k)*binomial(a2*n+b2*k,k)*x^k, for 1<=a1,a2<=A, 0<=b1,b2<=A, 1<=x<=B, try:`):
print(`DELsipur1(1,1,k,n,1000):`):



elif nargs=1 and args[1]=DELsipur2 then
print(`DELsipur2(A,B,k,n,K): outputs an article for DEL(F,k,n,K) for all F of the form binomial(n,k)*binomial(a1*n+b1*k,k)*binomial(a2*n+b2*k,k)*binomial(a3*n+b3*k,k)x^k, for 1<=a1,a2,a3<=A, 0<=b1,b2,b3<=A, 1<=x<=B, try:`):
print(`DELsipur2(1,1,k,n,1000):`):


elif nargs=1 and args[1]=DELsipur3 then
print(`DELsipur3(n,k,K): outputs an article for DEL(F,k,n,K) for all F in the pre-computed OutF1(n,k). Try:`):
print(`DELsipur3(n,k,K)`):

elif nargs=1 and args[1]=DELv then
print(`DELv(F,k,n,K): a verbose version of DEL(F,k,n,K) (q.v.). Try: `):
print(``):
print(`DELv(binomial(n,k)*binomial(n+k,k)*binomial(n+2*k,k),k,n,1000):`):
print(`DELv(binomial(n,k)*binomial(n+k,k)*binomial(n+2*k,k)*a^k,k,n,1000):, for a=2,3,4,`):
print(`DELv(binomial(n,k)*binomial(n+2*k,k)*a^k,k,n,1000):, for a=3,4,`):
print(` DELv(binomial(n,k)*binomial(2*n+2*k,2*k)*3^k,k,n,1000):`):
print(`DELv(binomial(n-k,2*k)*a^(n-k),k,n,1000):, a=1,2,3,...`):
print(`DELv(binomial(n,k)*binomial(n+k,k)^2*a^k,k,n,1000):, a=1,2,3,...`):
print(` DELv(binomial(n,k)^2*binomial(n+2*k,2*k)*a^k,k,n,1000):, a=1,2,3,4,...`):
print(` DELv(binomial(n,k)^3*binomial(n+k,k)^2*a^k,k,n,1000):, a=1,2,3...`):



elif nargs=1 and args[1]=DELazG then
print(`DELazG(F,x,n,K): inputs a hyperexponential function F (cont. in x, discrete in n) in the cont. variable x and the discrete variable n`):
print(`and outputs  a(K)/b(K) where a(n),b(n) are solutions of the recurrence satisfied by the controur integral of F around the origin`):
print(`with initial conditions [1,0,...0] and [0,1,...] respectively. It also gives you the estimated delta of the sequence. Try:`):
print(`DELazG((1/x+1+x)^n/x,x,n,1000);`):
print(`a:=17; DELazG(((1+a*x)*(1+(a+1)*x)/x)^n/x*x^(1/3),x,n,1000); solve(64+144*x*(1+2*a)+108*x^2*(3*a^2+3*a+1)+27*x^3*(1+2*a),x)[1];`):
print(`b:=17; DELazG(((1+b*x)*(1+(b+1)*x)/x)^n/x*x^(1/2),x,n,1000); -3*b-3/2-3*(b^2+b)^(1/2); `):

elif nargs=1 and args[1]=DELazGv then
print(`DELazGv(F,x,n,K):  a verbose form of DELazG(F,x,n,K) (q.v.)`):
print(`Try:`):
print(`a:=17; DELazGv(((1+a*x)*(1+(a+1)*x)/x)^n/x*x^(1/3),x,n,1000); solve(64+144*x*(1+2*a)+108*x^2*(3*a^2+3*a+1)+27*x^3*(1+2*a),x)[1];`):
print(`b:=17; DELazGv(((1+b*x)*(1+(b+1)*x)/x)^n/x*x^(1/2),x,n,1000); -3*b-3/2-3*(b^2+b)^(1/2); `):

elif nargs=1 and args[1]=DELmaz then
print(`DELmaz(a,b,c,K): inputs positive integers a,b,c, and outputs the Apery limit for the operaor OpeMAZ1(a,b,c,n,N) (q.v.( wuth initial conditions a(1)=1 and the rest 0, b(0)=1 and the rest 0. Try:`):
print(`DELmaz(1,1,1,1000);`):


elif nargs=1 and args[1]=FindAZ1 then
print(`FindAZ1(x,A,K): Inputs a positive integer d, a variable x, a positive integer A and a large positive integer K`):
print(`outputs the list of pairs [P,delta] such that P is a Laurent polynomial of low-degree -1, degree 1, and the coefficients`):
print(`are positive integers between 1 and A, and the implied delta (using s sequence of length 2*K), in the Apery limit of the recurrence`):
print(`satisfied by the constant term of P^n  yields a positive delta, followed by a conjectured value (produced by`):
print(`Maple, identified or own, if known. Try:`):
print(`FindAZ1(x,3,300);`):

elif nargs=1 and args[1]=FindAZ2 then
print(`FindAZ2(x,A,K): Inputs a positive integer d, a variable x, a positive integer A and a large positive integer K`):
print(`outputs the list of pairs [P,delta] such that P is a Laurent polynomial of low-degree -2, degree 2, and the coefficients`):
print(`are positive integers between 1 and A, and the implied delta (using s sequence of length 2*K), in the Apery limit of the recurrence`):
print(`satisfied by the constant term of P^n  yields a positive delta, followed by a conjectured value (produced by`):
print(`Maple, identified or own, if known. Try:`):
print(`FindAZ2(x,3,300);`):


elif nargs=1 and args[1]=FindMAZ then
print(`FindMAZ(A,K): Inputs a positive  integer A and a large positive integer K`):
print(`outputs the list of pairs [[a,b,c],DELmaz(a,b,c,K)] such that DELmaz(a,b,c,K)[2] is positive`):
print(`for all 1<=a,b,c,<=A. Try`):
print(`FindMAZ(1,1000);`):

elif nargs=1 and args[1]=FindZ1 then
print(`FindZ1(A,n,k,K): Inputs a positive  integer A and a large positive integer K`):
print(`outputs the list of pairs [F,DEL(F,k,n,K)] such that F is an expression of the form binomial(n,k)*binomial(a1*n+b1*k,a2*n+b2*k)*x^k`):
print(`where a1 is from 1 to A and b1 from 0 to A and a2<=a1 and b2<=b1 and x is between 1 and A`):
print(`such that DEL(F,k,n,K)[2] is positive. Try:`):
print(`FindZ1(1,n,k,300);`):

elif nargs=1 and args[1]=FindZ2 then
print(`FindZ2(A,n,k,K): Inputs a positive  integer A and a large positive integer K`):
print(`outputs the list of pairs [F,DEL(F,k,n,K)] such that F is an expression of the form binomial(n,k)*binomial(a1*n+b1*k,k)*binomial(a2*n+b2*k,k)*x^k`):
print(`where a1 is from 1 to A and b1 from 0 to A and a2<=a1 and b2<=b1 and x is between 1 and A`):
print(`such that DEL(F,k,n,K)[2] is positive. Try:`):
print(`FindZ2(1,n,k,300);`):

elif nargs=1 and args[1]=GEk then
print(`GEk(k,a,x): Guesses a polynomial equation in x, for numeric a (pos. integer) and pos. integer k>1, for DELazG(((1 + a * x) * (1 + (a + 1) * x) / x)^n / x^(1 / k), x, n, 1000)[1]:`):
print(`Try: GEk(3,13,x);`):

elif nargs=1 and args[1]=guessGEk then
print(`guessGEk(k,a,N): Guesses polynomial expressions for the coefficients of the polynomials GEk(k, a, x) using terms a = 1..N.`):
print(`Try: guessGEk(3,a,3);`):

elif nargs=1 and args[1]=guessPol then
print(`guessPol(L, x): Guesses a polynomial of minimal degree that generates the list L.`):
print(`Try: guessPol([1, 4, 9, 16, 25], n):`):

elif nargs=1 and args[1]=MyIDs then
print(`MyIDs(C,F,F0,N): Given a constant C in decimals and another constant`):
print(`let's call it F, whose floating-point if F0`):
print(` and a positive integer N`):
print(`tries to express C as (a*F+b)/(c*F+d) for a,b,c,d from -N to N using PSLQ`):
print(`MyIDs(evalf((log(2)-2)/(2*log(2)+3)),log(2),evalf(log(2)),100);`):

elif nargs=1 and args[1]=MyIDsLog then
print(`MyIDsLog(C,K,N): guesses a fractional-linear  expression for the constant C with coefficients <=N in aba. value (given in floating point) in terms of log(j/i) where 2<=i<j<=K. Try:`):
print(`MyIDsLog( evalf( (log(5)-log(2))/(log(5)+log(2))) ,5,1000);`):

elif nargs=1 and args[1]=OperAZd then
print(`OperAZd(F,x,n,N): Same as AZd(F,x,n,N)[1] but an operator, ope, in decreasing powers of N, such that it is equivalent to 1-ope`):

elif nops([args])=1 and op(1,[args])=OpeMAZ1 then
print(`OpeMAZ1(a,b,c,n,N): The linear recurrence operator in n and the shift operator N satisfied by the constant term of (in z, w) of`):
print(`((1+z)*(1+w)*(a*w+b*z+c*w*z)/w/z)^n. Try:`):
print(`OpeMAZ1(1,1,1,n,N);`):


elif nargs=1 and args[1]=OperZeil then
print(`OperZeil(F,k,n,N): Same as zeil(F,k,n,N)[1] but an operator, ope, in decreasing powers of N, such that it is equivalent to 1-ope`):


elif nargs=1 and args[1]=OutAZ1 then
print(`OutAZ1(x): The pre-computed output for  FindAZ1(x,30,500). Try:`):
print(`OutAZ1(x):`):


elif nargs=1 and args[1]=OutF1 then
print(`OutF1(n,k): The pre-computed output for  FindZ1(3,n,k,500). Try:`):
print(`OutF1(n,k):`):


elif nargs=1 and args[1]=QuadStory then
print(`QuadStory(c): An article about conjectured fast approximations to the root of  -3*c-3/2-3*(c^2+c)^(1/2)`):
print(`QuadStory(c):`):

elif nargs=1 and args[1]=RAT then
print(`RAT(F,k,n,K): inputs a closed-form expression and outputs  a(K)/b(K) where`):
print(`b(n):=Sum(F,k=0..n) and a(n) is satisfies the same initial conditions  with a(i)=0  for negative i and 0 and a(1)=1, Try`):
print(`RAT(binomial(n,k)^3,k,n,1000);`):
print(`RAT(binomial(n,k)^4,k,n,1000);`):
print(`RAT(binomial(n,k)*binomial(n+k,k)*binomial(n+2*k,k),k,n,1000);`):
print(`RAT(binomial(n,k)*binomial(n+k,k)*binomial(n+2*k,k)*a^k,k,n,1000), for a=2,3,4,`):

elif nargs=1 and args[1]=RATaz then
print(`RATaz(P,x,K): inputs a Laurent polynoimal P in the cont. variable x and the discrete variable n`):
print(`and outputs  a(K)/b(K) where`):
print(`b(n):=CT(P^n,x) and a(n) is satisfies the same initial conditions  with a(i)=0  for negative i and 0 and a(1)=1, Try`):
print(`RATaz(1/x+1+x,x,1000);`):


elif nargs=1 and args[1]=SeqFromRec then
print(`SeqFromRec(ope,n,N,Ini,n0,K): inputs an operator given in temrs of the negative shift operator`):
print(`1/N, and a list Ini, givining the values at n=n0, ...,n=n0+L-1, finds the list from n=1 to`):
print(`to n=K. Try:`):
print(`SeqFromRec(1/N+1/N^2,n,N,[0,1],0,20);`):

else
 print(`There is no such thing as`, args):

fi:


end:




##start: pre-computed data
#OutAZ1(x): The pre-computed output for  FindAZ1(x,30,500). Try:
#OutAZ1(x):
OutAZ1:=proc(x):
[[(x+1)*(2*x+1)/x, 1/2*ln(2), .290138663918575890477027023311636131963681884705\
021233477222579827306399380030243475337176562233842], [(3*x^2+3*x+1)/x, 1/18*3^
(1/2)*Pi, .14702545119145398375637667447861085665236783312456673338202690463398\
4122142782439956779794236138781], [(5*x^2+5*x+1)/x, .21520447048200201944471661\
64753027127122853412701448273785030519962806077305659806807451323486099777, .18\
6212997916515322508612740245897229868046913684172885908619261747900158676695576\
428908719407016826], [(2*x+1)*(3*x+1)/x, 1/2*ln(3)-1/2*ln(2), .4012379651122694\
1509544905969371551852527321235522925232170123821709829900077072177460490044787\
8509], [(7*x^2+5*x+1)/x, .19253082576711371276367574726314191299857385545021425\
28906121308502555303487174962331829263926134182, .78080073491229405823561304846\
895912562042516258775504751835989216306772848136316562359275472705757e-1], [(2*
x+1)*(4*x+1)/x, 1/4*ln(2), .289993055263313801565033684313653326472027841502219\
081584942200694592126617027277177787315087857137], [(10*x^2+6*x+1)/x, arcsin(1/
10*(50-15*10^(1/2))^(1/2)), .16636176530699067936987381578581868618761421474539\
3999079428821793745488166712126840476884743348785], [(12*x^2+6*x+1)/x, 1/36*3^(
1/2)*Pi, .146937495773266378857319005122457842950349455953808635382932008519215\
584482251022970797396371953119], [(11*x^2+7*x+1)/x, .14803728593913396657830557\
41315626073369158088368526324908891562031956524126983486401685289705691292, .10\
5116032640174596111076361042171348670638909600632511479206236185125878736630969\
33582133965855560e-1], [(3*x+1)*(4*x+1)/x, ln(2)-1/2*ln(3), .453196548180499242\
7664504006860637207239139886132684024998918064528064237416087495015744496082303\
67], [(13*x^2+7*x+1)/x, .140044318004706915480613604002123463987083950708263920\
5696104650795347482680550627105973941674569784, .161638335250753645026781872959\
954695956885975632142808209207093641315448080736429124433588858460764], [(14*x^
2+7*x+1)/x, .136583934550258933464647696017326394432270442825950707538107215965\
6904808804054835183669967750559749, .270373874926566062359375010151429119410368\
670304778779244283963217324477762282236609361271143588680], [(14*x^2+8*x+1)/x,
.130637614345119969946524659009519782933615647050865564314820980170892124192834\
9413329767703089758148, .129174561691876889204679137889893960139440257993731281\
53423581571285905401690360494547749960750308e-1], [(3*x+1)*(5*x+1)/x, 1/4*ln(5)
-1/4*ln(3), .110332623834382429793981533968855564523332788666435822604603966865\
375477075091165339727785921689243], [(17*x^2+8*x+1)/x, arcsin(1/34*(578-136*17^
(1/2))^(1/2)), .117293310412887240984508996420806444324999927618692529914917812\
513771309855960280624967674500211693], [(18*x^2+8*x+1)/x, .12015049158624418214\
0207472503093151608016240894608057167948397206419489338247930398089085368430607\
7, .311959701788433959980601829381404452205558290059324443634890645132630220925\
82843698636218067487320e-1], [(3*x+1)*(6*x+1)/x, 1/6*ln(2), .289907947391334325\
1690806888284194241545655451025540437684068273576963667599471665803675725178379\
24], [(19*x^2+9*x+1)/x, .113485945690054915812768267267329742235955492307352965\
1794418770288163207405405428935012016224164943, .773967828910504772723693597204\
15249577133527344675306198877377252766464299659640145237851677779470e-1], [(4*x
+1)*(5*x+1)/x, 1/2*ln(5)-ln(2), .4922845446468870964490032999148543427758089903\
23387458569092811456710730912867984778074746765201413], [(21*x^2+9*x+1)/x, .109\
7690682719225956686706290105507090487705192319091763710353583810983331723414843\
808379049760632766, .4067089156681885987799500616391069670896443895129287935554\
16967552887736025722207688866666755003907], [(27*x^2+9*x+1)/x, 1/54*3^(1/2)*Pi,
.147312182561576448842951589401225316071750348240331818734856976525758962788938\
832361711338140101166], [(20*x^2+10*x+1)/x, .1076022352410010097223583082376513\
563561426706350724136892515259981403038652829903403725661743049889, .1861077128\
1578328595159685356739800376444526743426988595181768762216223455113603905763991\
9883681372], [(4*x+1)*(6*x+1)/x, 1/4*ln(3)-1/4*ln(2), .401069804518428848148971\
268294478421760032230258576443828996747342716264790301092144048538097935088], [
(26*x^2+10*x+1)/x, arcsin(1/26*(338-65*26^(1/2))^(1/2)), .275457747700704031842\
008444477555257795663636733006946832023299341005999912766546346245178119370217]
, [(28*x^2+10*x+1)/x, .96265412883556856381837873631570956499286927725107126445\
30606542512776517435874811659146319630670909e-1, .78047690415448015914546490172\
093301555543925197667408646618947756524624241653832378757683996435603e-1], [(30
*x^2+10*x+1)/x, .94034336056763435181686002774047825079292973112947334175453858\
60544677198381079303252695013504968627e-1, .81348192803195195496240778497057268\
366951541458000405920199283589780106255471127858346287757085261e-1], [(29*x^2+
11*x+1)/x, .9219327514691665863321916315870214081535549697550627500377733115791\
949917861799563754029656875635181e-1, .1295868257311321969920735006168594061148\
54012364468715075663929318636189816589842966966171119948996], [(5*x+1)*(6*x+1)/
x, 1/2*ln(2)+1/2*ln(3)-1/2*ln(5), .51733600532054032820902906015003611956740829\
9277193212650479371298042172492604815842499988030901601], [(x+1)*(3*x+2)/x, 1/2
*ln(3)-1/2*ln(2), .401237965112269415095449059693715518525273212355229252321701\
238217098299000770721774604900447878509], [(5*x^2+6*x+2)/x, arcsin(1/10*(50-15*
10^(1/2))^(1/2)), .166361765306990679369873815785818686187614214745393999079428\
821793745488166712126840476884743348785], [(3*x+2)*(2*x+1)/x, ln(2)-1/2*ln(3),
.453196548180499242766450400686063720723913988613268402499891806452806423741608\
749501574449608230367], [(7*x^2+7*x+2)/x, .136583934550258933464647696017326394\
4322704428259507075381072159656904808804054835183669967750559749, .270373874926\
5660623593750101514291194103686703047787792442839632173244777622822366093612711\
43588680], [(7*x^2+8*x+2)/x, .1306376143451199699465246590095197829336156470508\
655643148209801708921241928349413329767703089758148, .1291745616918768892046791\
3788989396013944025799373128153423581571285905401690360494547749960750308e-1],
[(9*x^2+8*x+2)/x, .120150491586244182140207472503093151608016240894608057167948\
3972064194893382479303980890853684306077, .311959701788433959980601829381404452\
20555829005932444363489064513263022092582843698636218067487320e-1], [(3*x+2)*(3
*x+1)/x, 1/6*ln(2), .2899079473913343251690806888284194241545655451025540437684\
06827357696366759947166580367572517837924], [(2*x+1)*(5*x+2)/x, 1/2*ln(5)-ln(2)
, .4922845446468870964490032999148543427758089903233874585690928114567107309128\
67984778074746765201413], [(13*x^2+10*x+2)/x, arcsin(1/26*(338-65*26^(1/2))^(1/
2)), .2754577477007040318420084444775552577956636367330069468320232993410059999\
12766546346245178119370217], [(15*x^2+10*x+2)/x, .94034336056763435181686002774\
04782507929297311294733417545385860544677198381079303252695013504968627e-1, .81\
3481928031951954962407784970572683669515414580004059201992835897801062554711278\
58346287757085261e-1], [(5*x+2)*(3*x+1)/x, 1/2*ln(2)+1/2*ln(3)-1/2*ln(5), .5173\
3600532054032820902906015003611956740829927719321265047937129804217249260481584\
2499988030901601], [(16*x^2+11*x+2)/x, .892144668894316929309258311119504571651\
8487104723565893293488689922295893499162985078998393501845002e-1, .456802586661\
8051437005297036734428249143875043631897718073704246369884686408723678372781950\
8189198e-1], [(17*x^2+12*x+2)/x, .849301172328479520523317629210206305907090091\
2711015983375237981685523117059978030387807506822794549e-1, .122406710560326424\
1305230545093989239679816348846807481121239098264412988980896928000397227495105\
03], [(19*x^2+12*x+2)/x, .81839606877249865641136752645931304487764419899674008\
49580847995465457184899226703439723968535278178e-1, .13423819575066554404039370\
8511513711018662665820576784119058239041785367865310047442313149984357620], [(5
*x+2)*(4*x+1)/x, 1/2*ln(2)-1/6*ln(5), .1967371220351482494835541771147908682718\
3586064566029123956224194553758281227366662910045527383015e-1], [(3*x+1)*(7*x+2
)/x, 1/2*ln(7)-1/2*ln(2)-1/2*ln(3), .540325606383608762463721000811538373984477\
419396947784472402781742008087593173395152460371570840177], [(22*x^2+13*x+2)/x,
.758866602942512619466779295799035195463555206291091963629401434083257727957352\
7290819229322993544913e-1, .833116135862443758645752676901837601797151062907720\
36798792592256932587783291185138553793248516451e-1], [(21*x^2+14*x+2)/x, .75154\
9718503086111183746818025502768644978435227740139001082750536772066769397420046\
3726324464985163e-1, .106942675499504497258164464390882197948936840636641014524\
849049825538285145974043800504080483866547], [(23*x^2+14*x+2)/x, .7294230569649\
7496523053344255320495576079014552484796945674928538141651734550846619994814379\
67006056e-1, .60096436780636933495673629793571449225965072780997876780909072222\
698317200943519759713252611900038e-1], [(25*x^2+14*x+2)/x, arcsin(1/10*(50-35*2
^(1/2))^(1/2)), .33863462746087629647735512319390329176212776720308115047155324\
4938113635314771229758662072207763733], [(5*x+2)*(5*x+1)/x, 1/10*ln(2), .289800\
7952403449079130039846733641996318148661956812924898554856777678072490631203581\
47481380430566], [(3*x+1)*(9*x+2)/x, 1/6*ln(3)-1/6*ln(2), .40097150220307421551\
0590589154375222678437563200732098915489115067424961835232837165307752489830507
], [(7*x+2)*(4*x+1)/x, 3/2*ln(2)-1/2*ln(7), .5544210962377009199035047025564965\
54165716433172164266221254095832361801972439915361067294714249702], [(29*x^2+15
*x+2)/x, .659879339736069396638023192081805215762895182608479683764652528768206\
8337583920453938307232029484360e-1, .115141797587130561899556765755314983937814\
763930779101748736221526101909642518799513429738909034979], [(30*x^2+15*x+2)/x,
.652417613387141093493285411349226549882232372004965119153293206019921447781760\
5188744976432300036974e-1, .355774981103955385866110905839630184247919789673403\
361251022997452397914726352730352552216799353213], [(4*x^2+6*x+3)/x, 1/36*3^(1/
2)*Pi, .14693749577326637885731900512245784295034945595380863538293200851921558\
4482251022970797396371953119], [(x+1)*(4*x+3)/x, ln(2)-1/2*ln(3), .453196548180\
4992427664504006860637207239139886132684024998918064528064237416087495015744496\
08230367], [(x+1)*(5*x+3)/x, 1/4*ln(5)-1/4*ln(3), .1103326238343824297939815339\
68855564523332788666435822604603966865375477075091165339727785921689243], [(6*x
^2+8*x+3)/x, .12015049158624418214020747250309315160801624089460805716794839720\
64194893382479303980890853684306077, .31195970178843395998060182938140445220555\
829005932444363489064513263022092582843698636218067487320e-1], [(7*x^2+9*x+3)/x
, .1097690682719225956686706290105507090487705192319091763710353583810983331723\
414843808379049760632766, .4067089156681885987799500616391069670896443895129287\
93555416967552887736025722207688866666755003907], [(4*x+3)*(2*x+1)/x, 1/4*ln(3)
-1/4*ln(2), .401069804518428848148971268294478421760032230258576443828996747342\
716264790301092144048538097935088], [(10*x^2+10*x+3)/x, .9403433605676343518168\
600277404782507929297311294733417545385860544677198381079303252695013504968627e\
-1, .81348192803195195496240778497057268366951541458000405920199283589780106255\
471127858346287757085261e-1], [(5*x+3)*(2*x+1)/x, 1/2*ln(2)+1/2*ln(3)-1/2*ln(5)
, .5173360053205403282090290601500361195674082992771932126504793712980421724926\
04815842499988030901601], [(11*x^2+11*x+3)/x, .88295417853761896782455109440011\
78006628494558224065441903839600220059171091478726508029336640113096e-1, .32582\
6656266948239106030982492181409814254515335745441756590669004751026993592687381\
113574045386170], [(11*x^2+12*x+3)/x, .8577125750874812237933268457539347784274\
556970909649446645428089781586169125884465150975517795928491e-1, .7776894586059\
4467004095739482051269509258986944760687499381511476555853503569306769300172090\
932864e-1], [(13*x^2+12*x+3)/x, .8112778801716469647586638613578457903013021198\
692975434601851207590955762650195895171171860060985819e-1, .8772576033932414169\
9563918697076891223788330897030253464459814779164852306277219498914363928206696\
e-1], [(16*x^2+12*x+3)/x, 1/72*3^(1/2)*Pi, .14684964552775836643046514258945861\
5152744211215851735223654856458297767122302721853518458231159877], [(13*x^2+13*
x+3)/x, .7899181796594889652723542155561019426013185926486066666293376718165684\
237124773386243062264950432773e-1, .3313319524085714477724781124274822437951460\
69371405191826386816057235821339783067476916052922893664], [(2*x+1)*(7*x+3)/x,
1/2*ln(7)-1/2*ln(2)-1/2*ln(3), .54032560638360876246372100081153837398447741939\
6947784472402781742008087593173395152460371570840177], [(14*x^2+14*x+3)/x, .751\
5497185030861111837468180255027686449784352277401390010827505367720667693974200\
463726324464985163e-1, .1069426754995044972581644643908821979489368406366410145\
24849049825538285145974043800504080483866547], [(2*x+1)*(8*x+3)/x, 1/2*ln(2)-1/
4*ln(3), .453024429709461714852038278981369484610531734653926368091980988907426\
832382862903412920981228501601], [(17*x^2+14*x+3)/x, .7047988118784926283706913\
599267386030688253675923505905708218358864090066808096237060356016387612910e-1,
.702344267848258879366165935958822677958207178700727405527168766544192772022107\
33663215581800807033e-1], [(19*x^2+15*x+3)/x, .66372718397370036441987514809608\
32860464328677720489495171413554157748475108558336488452799144086278e-1, .49175\
8757526156337585012186554335055143199561244551531107241266162321714869427433371\
257525484559750], [(20*x^2+15*x+3)/x, .6524176133871410934932854113492265498822\
323720049651191532932060199214477817605188744976432300036974e-1, .3557749811039\
5538586611090583963018424791978967340336125102299745239791472635273035255221679\
9353213], [(25*x^2+15*x+3)/x, 1/90*3^(1/2)*Pi, .1468213864134730076898156151821\
34735697507450222873227411337325389294248482408205444767250679623919], [(2*x+1)
*(10*x+3)/x, 1/8*ln(5)-1/8*ln(3), .11070235858245017727653434882085814322625560\
5113179734281693480681761469291069442006801117220572360], [(7*x+3)*(3*x+1)/x, 1
/2*ln(3)-1/4*ln(7), .2526575634031649995382606827058945838739236322766906233817\
32943815038977927792874503122472168613460], [(22*x^2+16*x+3)/x, .61860899343973\
6476616780596152149391139050621971779427211605550476200087835596326055877785425\
8289550e-1, .188906027419001087261899103035624131711333621835880083272117558797\
624075038795755009850129851498794], [(24*x^2+16*x+3)/x, .6007524579312209107010\
373625154657580400812044730402858397419860320974466912396519904454268421530384e\
-1, .31183412875274435017702934047519370120596783542896078659391433001416184373\
722807548354717396129118e-1], [(8*x+3)*(3*x+1)/x, ln(3)-3/2*ln(2), .56148143177\
2305369228401183798683261755051559503833850196854648011043608743060875618336232\
642393356], [(25*x^2+17*x+3)/x, .5809380401645286932845839538933593430322189548\
043015928749092971870836258588018482326301833941239337e-1, .3855936619868289360\
5538947497514400946418454818720473501153611100234492745824113523241532894825666\
e-1], [(26*x^2+18*x+3)/x, .5625708338586260381578117769348148903118350424572836\
423420400122983851411098925064855857658563755154e-1, .2813666621260850279530985\
90523765884567027656884767418652472220015980453213766102707810423847536191], [(
28*x^2+18*x+3)/x, .548845341359612978343353145052753545243852596159545881855176\
7919054916658617074219041895248803163831e-1, .406541379119630588444842881897470\
533786290398688027187509360575469566544136461156319309310510099007], [(29*x^2+
18*x+3)/x, .5424172927286820168242345115236634533960267583759247780555379289951\
543625562525999492088353881231129e-1, .2309383894973826571615649396990277464496\
102361940850326000197012669234311276568910291704293792617e-2], [(29*x^2+19*x+3)
/x, .53277364045104324039856273722364387613403875557388297000566313605240626509\
06679652228965952352612634e-1, .22497035742215844153790725266835149589721894655\
024077907297388372336381759339132184664557656039675e-1], [(3*x+1)*(10*x+3)/x, 1
/2*ln(2)+1/2*ln(5)-ln(3), .5761692654119481969663845581964874622941936093538875\
48607469536138205891433841239666613502589801178], [(30*x^2+20*x+3)/x, .51774414\
7457030956262558168189715860006407377317055245025568165474594368905128461809766\
4366479131960e-1, .455698600138876218199221504538339663678649551550226329268695\
14822591164629366212217603649657616167e-1], [(x+1)*(5*x+4)/x, 1/2*ln(5)-ln(2),
.492284544646887096449003299914854342775808990323387458569092811456710730912867\
984778074746765201413], [(5*x^2+10*x+4)/x, .10760223524100100972235830823765135\
63561426706350724136892515259981403038652829903403725661743049889, .18610771281\
5783285951596853567398003764445267434269885951817687622162234551136039057639919\
883681372], [(7*x^2+10*x+4)/x, .96265412883556856381837873631570956499286927725\
10712644530606542512776517435874811659146319630670909e-1, .78047690415448015914\
546490172093301555543925197667408646618947756524624241653832378757683996435603e\
-1], [(8*x^2+11*x+4)/x, .892144668894316929309258311119504571651848710472356589\
3293488689922295893499162985078998393501845002e-1, .456802586661805143700529703\
67344282491438750436318977180737042463698846864087236783727819508189198e-1], [(
5*x+4)*(2*x+1)/x, 1/2*ln(2)-1/6*ln(5), .196737122035148249483554177114790868271\
83586064566029123956224194553758281227366662910045527383015e-1], [(11*x^2+13*x+
4)/x, .758866602942512619466779295799035195463555206291091963629401434083257727\
9573527290819229322993544913e-1, .833116135862443758645752676901837601797151062\
90772036798792592256932587783291185138553793248516451e-1], [(11*x^2+14*x+4)/x,
.740186429695669832891527870657813036684579044184263162454445781015978262063491\
7432008426448528456462e-1, .105075756282448891464911637061625625541221421532947\
07106841946198519364978776266055724344882686624e-1], [(13*x^2+14*x+4)/x, .70022\
1590023534577403068020010617319935419753541319602848052325397673741340275313552\
9869708372848921e-1, .161577177301562588541916597599286194069083688607012352769\
012712324502612040351602374905809480780710], [(7*x+4)*(2*x+1)/x, 3/2*ln(2)-1/2*
ln(7), .55442109623770091990350470255649655416571643317216426622125409583236180\
1972439915361067294714249702], [(15*x^2+15*x+4)/x, .652417613387141093493285411\
3492265498822323720049651191532932060199214477817605188744976432300036974e-1, .\
3557749811039553858661109058396301842479197896734033612510229974523979147263527\
30352552216799353213], [(3*x+2)*(5*x+2)/x, 1/8*ln(5)-1/8*ln(3), .11070235858245\
0177276534348820858143226255605113179734281693480681761469291069442006801117220\
572360], [(17*x^2+16*x+4)/x, arctan(RootOf(_Z^4+16*_Z^3-6*_Z^2-16*_Z+1,index =
1)), .1176605349067828112001458888745170180265966656170936236564558138901858434\
52198711055128873831844648], [(17*x^2+17*x+4)/x, .60019432926895200993834890980\
00372511696421714199915636554428250224777596076376507533428748823867607e-1, .35\
7071079319350908159085047237246230274950970742020673222119646805254269077961916\
037376091849120324], [(2*x+1)*(9*x+4)/x, ln(3)-3/2*ln(2), .56148143177230536922\
8401183798683261755051559503833850196854648011043608743060875618336232642393356
], [(19*x^2+18*x+4)/x, .5674297284502745790638413363366487111797774615367648258\
972093851440816037027027144675060081120824713e-1, .7736916761770549453182318137\
6467416781267247343424567310803561996890124825607455393030568444283870e-1], [(
21*x^2+18*x+4)/x, .548845341359612978343353145052753545243852596159545881855176\
7919054916658617074219041895248803163831e-1, .406541379119630588444842881897470\
533786290398688027187509360575469566544136461156319309310510099007], [(27*x^2+
18*x+4)/x, 1/108*3^(1/2)*Pi, .1472242804677610330017446167668096083757434536604\
48702710036620696303459769626327032835987300197282], [(19*x^2+19*x+4)/x, .55697\
8652460532651525150971862570754468868189373160799626783507694577434438292032210\
9965172002084211e-1, .172462820676702089641539660234619029133781035395653466971\
77178657436689273478276357002495054710705e-1], [(2*x+1)*(11*x+4)/x, 1/6*ln(11)-\
1/2*ln(2), .1042756835772791772249920742337847164603293397929152921789286869098\
32876423992406695525927389174440], [(23*x^2+19*x+4)/x, .52295297137846159532693\
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294904850], [(7*x+5)*(3*x+2)/x, 1/2*ln(3)+1/2*ln(5)-1/2*ln(2)-1/2*ln(7), .61025\
8257038192286759439974027672515980347175576021590227862119309511474800841355626\
763630280091485], [(21*x^2+30*x+10)/x, .341051920274759928528928386632467785723\
0240163234248312183066455215426010342716356232240718469065848e-1, .218337782091\
9457385181735669932769983780252002949968093592406323492442623643532930499949236\
51787441], [(23*x^2+30*x+10)/x, .3308966056439973332320641668126660804476898793\
168568746943276811334298734316153430255377745843493225e-1, .3307376908241285861\
0180053745224081256487632895625874440661993464188316985547808289021507647257833\
1], [(27*x^2+30*x+10)/x, .31344778685587811727228667591349275026430991037649111\
39181795286848225732793693101084231671168322876e-1, .81286363473735685565819051\
599751242973133916046744324061244150915974150095029841136786867610504624e-1], [
(12*x^2+22*x+11)/x, .4414770892688094839122755472000589003314247279112032720951\
919800110029585545739363254014668320056548e-1, .3256716933764311082429898480740\
64679179002561449196555200986125684336160503514051013736248884401313], [(x+1)*(
12*x+11)/x, ln(2)+1/2*ln(3)-1/2*ln(11), .59038482086041270771311794057430065523\
9352045485774545913177958159480443227414084269321301413668508], [(12*x^2+24*x+
11)/x, .42885628754374061189666342287696738921372784854548247233227140448907930\
84562942232575487758897964246e-1, .77738636927006377536253998752851379086183007\
711978977988615673519154709167974846626446098690284695e-1], [(x+1)*(13*x+11)/x,
1/4*ln(13)-1/4*ln(11), .3182033387315036414980420781180503789324000902950468092\
69597133682451258631957243718126047125848140], [(x+1)*(14*x+11)/x, 1/6*ln(2)+1/
6*ln(7)-1/6*ln(11), .1532490804951133228519620923282424614377785984124317201961\
10628999102577311822285927735596625874798], [(15*x^2+25*x+11)/x, .3927745961004\
9893260899868256221905952317311053872629808293931392192343129601086345345987086\
17937180e-1, .49684500773734013813873242167201989076668816518744208286851504217\
797059057573167267535616138857182e-1], [(x+1)*(15*x+11)/x, 1/8*ln(3)+1/8*ln(5)-\
1/8*ln(11), .436512359208337077352427737024203908761458376470264962420875787316\
31109072010985671281805536452538e-1], [(16*x^2+26*x+11)/x, .3794333014712563097\
3338964789951759773177760314554598181470071704162886397867636454096146614967724\
56e-1, .83284608990407182045595032496367722007605216405670120220542528920050567\
834299379329740232465093405e-1], [(17*x^2+27*x+11)/x, .367202102966508167230121\
5468855480489632763517956081441463021613029875019837407880873091686149621604e-1
, .1522545226824009973477268735375107551105245105161625131596257622100964454120\
1785661415958428650702e-1], [(16*x^2+28*x+11)/x, .37009321484783491644576393532\
89065183422895220921315812272228905079891310317458716004213224264228231e-1, .10\
5035510777555643715254608093421856518109925167203712104445275867785227025246250\
03567086452619466e-1], [(18*x^2+28*x+11)/x, .3559354691155796956650329350893868\
963748443204689061046975407457752793177778559120823652682756902032e-1, .2856193\
3568329605758962889796536415335086128792061270884020835643858076844748149863283\
3270053878864], [(19*x^2+29*x+11)/x, .34551340249079050765537306864232865100960\
31652949966731144727917437933167215780574666732734815263498e-1, .29584269119229\
2759842186781259983505795233217055476775840710740800485779735212682988251422350\
433894], [(20*x^2+30*x+11)/x, .335835922714340264332055211718700526876847662166\
4609744380694789654247765893381602028830168902042423e-1, .454110681358296829795\
798163130505598435824431885544171232331727531135582787420084270458193150965781]
, [(21*x^2+30*x+11)/x, .3304168931087764128548778239898299520056922995433342143\
962716676342898443844261790116835371694318096e-1, .1160718868642284509271998426\
95535623866793903670880125431967748084270026380185124384684436381338602], [(13*
x^2+24*x+12)/x, .40563894008582348237933193067892289515065105993464877173009256\
03795477881325097947585585930030492910e-1, .87692324920564651648254284545978615\
826325176937052470495534275017436138941474882609083330931944101e-1], [(x+1)*(13
*x+12)/x, 1/2*ln(13)-ln(2)-1/2*ln(3), .5984780616229934616825182548810213832843\
04362272684235354091641939306635125346450560445818929722175], [(13*x^2+26*x+12)
/x, .39495908982974448263617710777805097130065929632430333331466883590828421185\
62386693121531132475216387e-1, .33117740418293312688211778725135590352338462153\
1596008557544606834961360263756042057223119127565863], [(17*x^2+28*x+12)/x, .35\
2399405939246314185345679963369301534412683796175295285410917943204503340404811\
8530178008193806455e-1, .702118275353624079933266089650323227402815826021950722\
39554686620157521585343508519424531088728917e-1], [(19*x^2+30*x+12)/x, .3318635\
9198685018220993757404804164302321643388602447475857067770788742375542791682442\
26399572043139e-1, .49158193295376732395659223254486326758985523623621270258319\
9293381034941439321252116180444741959377], [(25*x^2+30*x+12)/x, 1/180*3^(1/2)*
Pi, .14673367490942376821290997702267427404476001653868147483569873154298287108\
6561190168875228864038911], [(14*x^2+26*x+13)/x, .37518501653225902526981292145\
50130546753619616645005977428264638936114568751259475342993192572040525e-1, .20\
7144406522972963229093759337998211847134275888595219892544067312123052199726493\
391970315343397058], [(x+1)*(14*x+13)/x, 1/2*ln(2)+1/2*ln(7)-1/2*ln(13), .60259\
5596472095888421942266891565516351873903990545910111868259519470889156357283389\
181872220371035], [(14*x^2+28*x+13)/x, .366030380660590013307513764208033600542\
6892669479481601059981887942434063514341334867673600291314363e-1, .945644856516\
0695757719213750580822732395653257471296639430386558975209289769956801441757472\
7666799e-1], [(x+1)*(15*x+13)/x, 1/4*ln(3)+1/4*ln(5)-1/4*ln(13), .3390235418698\
8147902692705187457309992953740087457829951957243003234304808769616548999656898\
4576430], [(16*x^2+28*x+13)/x, .35011079501176728870153401000530865996770987677\
06598014240261626988368706701376567764934854186424461e-1, .16151606561466555954\
5390648534945235498288747816624714583966491868462558272674386086096672621266388
], [(x+1)*(16*x+13)/x, 2/3*ln(2)-1/6*ln(13), .178361342906917675346896976049319\
219792313005274976607133331555750576647385604950761346176211191414], [(x+1)*(17
*x+13)/x, 1/8*ln(17)-1/8*ln(13), .722986035493143374760053941444379761393966549\
31642252509117610110776002038864451809118897525380342e-1], [(18*x^2+30*x+13)/x,
arctan(RootOf(_Z^6+30*_Z^5-15*_Z^4-100*_Z^3+15*_Z^2+30*_Z-1,index = 1)), .27529\
2715862140233563157652158582612984723723367891320884714033432874601263941724812\
081061647229479], [(15*x^2+28*x+14)/x, .348986269862877990150343289669640785630\
5315525689241589229888341007549137445117500163539028108535097e-1, .100770969991\
8617329140084375515815591977123778636762859690031700149452991388551100635946912\
94904850], [(x+1)*(15*x+14)/x, 1/2*ln(3)+1/2*ln(5)-1/2*ln(2)-1/2*ln(7), .610258\
2570381922867594399740276725159803471755760215902278621193095114748008413556267\
63630280091485], [(15*x^2+30*x+14)/x, .3410519202747599285289283866324677857230\
240163234248312183066455215426010342716356232240718469065848e-1, .2183377820919\
4573851817356699327699837802520029499680935924063234924426236435329304999492365\
1787441], [(16*x^2+30*x+15)/x, .32620880669357054674664270567461327494111618600\
24825595766466030099607238908802594372488216150018487e-1, .35561480163374955379\
2911060471768586984281430748898860030420393528682579245805286238264483255803609
]]:
end:



OutF1:=proc(n,k):
[[binomial(n,k)*binomial(n+k,n), [1/2*ln(2), .290138663918575890477027023311636\
131963681884705021233477222579827306399380030243475337176562233842]], [binomial
(n,k)*binomial(n+k,n)*2^k, [1/2*ln(3)-1/2*ln(2), .40123796511226941509544905969\
3715518525273212355229252321701238217098299000770721774604900447878509]], [
binomial(n,k)*binomial(n+k,n)*3^k, [ln(2)-1/2*ln(3), .4531965481804992427664504\
00686063720723913988613268402499891806452806423741608749501574449608230367]], [
binomial(n,k)*binomial(n+2*k,n)*3^k, [1/54*3^(1/2)*Pi, .71699820587329741284720\
735017541101386666958791818915689561735532283572055932292243894020611470279e-1]
], [binomial(n,k)*binomial(2*n+k,n), [2/7*ln(2), .10092997549010473425745664083\
9929004037476828304350759201740738902460510622601288355767806812824965]], [
binomial(n,k)*binomial(2*n+k,n)*2^k, [4/13*ln(3)-4/13*ln(2), .16205826831482143\
3494741540010055747846931263046813264506669618057192172659244894614946969028743\
963]], [binomial(n,k)*binomial(2*n+k,n)*3^k, [12/19*ln(2)-6/19*ln(3), .19692403\
3239225634033511528765316279176601379898409295075300719486090912060662723919938\
623638473649]], [binomial(n,k)*binomial(2*n+k,n+k), [2/7*ln(2), .10092997549010\
4734257456640839929004037476828304350759201740738902460510622601288355767806812\
824965]], [binomial(n,k)*binomial(2*n+k,n+k)*2^k, [4/13*ln(3)-4/13*ln(2), .1620\
5826831482143349474154001005574784693126304681326450666961805719217265924489461\
4946969028743963]], [binomial(n,k)*binomial(2*n+k,n+k)*3^k, [12/19*ln(2)-6/19*
ln(3), .19692403323922563403351152876531627917660137989840929507530071948609091\
2060662723919938623638473649]], [binomial(n,k)*binomial(2*n+k,2*n), [4/11*ln(2)
, .3963780639003273596297094195491622950657454406533966636396644440392599686488\
5570814172902825787745e-1]], [binomial(n,k)*binomial(2*n+k,2*n)*2^k, [6/17*ln(3
)-6/17*ln(2), .1347971356348744904092555937400976733959112387688475938343515775\
43313756224739429694813701271623131]], [binomial(n,k)*binomial(2*n+k,2*n)*3^k,
[16/23*ln(2)-8/23*ln(3), .17337475900378443717609164513798486336179972224262332\
2449810007071461352623334863429128370535809395]], [binomial(n,k)*binomial(2*n+2
*k,n), [1/4*ln(2), .27514814790996379039862114843236039696699031358072236059572\
135294218968905381651556064164641876981e-1]], [binomial(n,k)*binomial(2*n+2*k,n
)*2^k, [1/4*ln(3)-1/4*ln(2), .7229702398391232436407050856583857150792226809872\
9919436288279006846095352555722036593687460491010e-1]], [binomial(n,k)*binomial
(2*n+2*k,n)*3^k, [1/2*ln(2)-1/4*ln(3), .977251967264033458939801389394922061022\
07629696163339703504587401731059137644732026137177966274834e-1]], [binomial(n,k
)*binomial(2*n+2*k,n+2*k), [1/4*ln(2), .275148147909963790398621148432360396966\
99031358072236059572135294218968905381651556064164641876981e-1]], [binomial(n,k
)*binomial(2*n+2*k,n+2*k)*2^k, [1/4*ln(3)-1/4*ln(2), .7229702398391232436407050\
8565838571507922268098729919436288279006846095352555722036593687460491010e-1]],
[binomial(n,k)*binomial(2*n+2*k,n+2*k)*3^k, [1/2*ln(2)-1/4*ln(3), .977251967264\
0334589398013893949220610220762969616333970350458740173105913764473202613717796\
6274834e-1]], [binomial(n,k)*binomial(2*n+2*k,2*n)*3^k, [.525221631090500407427\
5805302065255858571534068864051837028967165876169841421110129102435875978287949\
e-1, .1163534013056112877307945986273336174881911501475415876535262532411676812\
39091107031951677330484385]], [binomial(n,k)*binomial(3*n+k,n)*2^k, [24/107*ln(
3)-24/107*ln(2), .3401228474162977341223750155893159109965545678864631184836644\
4970742197516919372395101177682512754e-1]], [binomial(n,k)*binomial(3*n+k,n)*3^
k, [108/233*ln(2)-54/233*ln(3), .6937226766502199597406500968403189493543489320\
1127382636016179611206184472057940357896475267184360e-1]], [binomial(n,k)*
binomial(3*n+k,n+k), [4/25*ln(2), .14548445135064214757017890396854199849870423\
3415639705869983196085914012052709769983035851892023651]], [binomial(n,k)*
binomial(3*n+k,n+k)*2^k, [12/73*ln(3)-12/73*ln(2), .185891465737021824266547600\
452475453705206616744238289524695722581918405297960789380405622413999480]], [
binomial(n,k)*binomial(3*n+k,n+k)*3^k, [48/145*ln(2)-24/145*ln(3), .20347559256\
5044320947271922075340798203183693593321144590921383820433867641447130609887347\
234507624]], [binomial(n,k)*binomial(3*n+k,2*n), [4/25*ln(2), .1454844513506421\
4757017890396854199849870423341563970586998319608591401205270976998303585189202\
3651]], [binomial(n,k)*binomial(3*n+k,2*n)*2^k, [12/73*ln(3)-12/73*ln(2), .1858\
9146573702182426654760045247545370520661674423828952469572258191840529796078938\
0405622413999480]], [binomial(n,k)*binomial(3*n+k,2*n)*3^k, [48/145*ln(2)-24/
145*ln(3), .2034755925650443209472719220753407982031836935933211445909213838204\
33867641447130609887347234507624]], [binomial(n,k)*binomial(3*n+k,2*n+k)*2^k, [
24/107*ln(3)-24/107*ln(2), .340122847416297734122375015589315910996554567886463\
11848366444970742197516919372395101177682512754e-1]], [binomial(n,k)*binomial(3
*n+k,2*n+k)*3^k, [108/233*ln(2)-54/233*ln(3), .69372267665021995974065009684031\
894935434893201127382636016179611206184472057940357896475267184360e-1]], [
binomial(n,k)*binomial(3*n+k,3*n)*2^k, [54/197*ln(3)-54/197*ln(2), .91127382062\
8910923399998324584402179236841585475262682655996771747847037338631940171284669\
5788768e-2]], [binomial(n,k)*binomial(3*n+k,3*n)*3^k, [192/359*ln(2)-96/359*ln(
3), .47001479141125815256589516844868357208318083273696887663550043811676318021\
037682144302886934310074e-1]]]:
end:

##End pre-computed data

Digits:=100:

####FROM EKHAD
#######################################################################
## EKHAD: Save this file as EKHAD. To use it, stay in the             #
## same directory, get into Maple (by typing: maple <Enter> )         #
## and then type:  read EKHAD : <Enter>                               #
## Then follow the instructions given there                           #
##                                                                    #
## Written by Doron Zeilberger, Rutgers University ,                  #
##  zeilberg@math.rutgers.edu.                                        # 
#######################################################################

with(SolveTools):
solve1:=Linear:
 
ezraEKHAD:=proc()
if args=NULL then
print(`EKHAD`):
print(`A Maple package for proving Hypergeometric (Binomial Coeff.)`):
print(` and other kinds of identities`):
print(`This version is for  Maple 8 (thanks to Drew Sills for noticing).`):
print(`The version (Feb, 25, 1999) is much faster than the previous`):
print(`version, thanks to a SLIGHT (yet POWERFUL) modification suggested by`):
print(` FREDERIC CHYZAK `):
print(``):
print(`For help with a specific procedure, type "ezra(procedure_name);"`):
print(`Contains procedures: `):
print(`findrec,ct,zeil,zeilpap,zeillim,AZd,AZdI, AZc,AZcI, AZpapd,AZpapc,celine, TerryTao `):
fi:
 
 
if nops([args])=1 and op(1,[args])=`celine` then
print(`celine(FUNCTION,ORDER_n,ORDER_k) applies Sister Celine's technique`):
print(`It inputs a function (n,k)->FUNCTION, and the guessed orders in`):
print(`n and k, ORDER_n,ORDER_k respectively, and outputs a partial`):
print(`k-free recurrence`):
print(`it  also finds the telescoped form of the recurrence.`):   
print(` In input, do not raise products of factorials to powers;  `):  
print(`instead raise each factorial to the power. `):
print(`For example:celine((n,k)->binomial(n,k),1,1);`):
print(`celine ((n,k)->n!*(2*k)!*(-2)^(n-k)/(k!^3*(n-k)!),2,2);`):
 
fi:
 
 
if nops([args])=1 and op(1,[args])=`findrec` then
print(`findrec(f,DEG,ORDER,n,N) finds empirically an ordi. linear recurrence`):
print(` with polynomial coeffs. The input is a sequence f given as a list`):
print(`STARTING at f[1],i.e. f[0] is not considered`):
print(` where DEG:=the maximal degree of the coefficients`):
print(`and ORDER:=the order of the recurrence.  The output is the operator`):
print(` in n and N, where N is the forward unit shift: Nf(n):=f(n+1).`):
print(`For example findrec([1,1,2,3,5,8,13,21,34],0,2,n,N) should yield`):
print(`N^2-N-1 , and findrec([1,2,5,14,42,132,429],1,1,m,M) should yield`):
print(`(m+2)*M-(4*m+2). If there is not enough data, you will get an`):
print(`an error message. If there is no operator, you would get 0`):
fi:
 
if nops([args])=1 and op(1,[args])=`AZpapc` then
print(`AZpapc(INTEGRAND,y,x) inputs a hypergeometric integrand`):
print(`in the continuous variables y and x (i.e. the logarithmic derivatives`):
print(`diff(INTEGRAND,x)/INTEGRAND and diff(INTEGRAND,y)/INTEGRAND are`):
print(`rational functions in x and y)`):
print(`and outputs a paper with a proof of the linear differential equation`):
print(`that the definite integral w.r.t. to y (which is a function of x)`):
print(`satisfies. It uses the method of`):
print(`Gert Almkvist and Doron Zeilberger, "The method of differentiating`):
print(`under the integral sign", J. Symbolic Computation 10(1990), 571-591.`):
print(``):
print(`It could be used to establish the diff. eq. of the `):
print(`classical orthogonal polynomials, when they are defined in terms`):
print(`or their generating funtion.`):
print(`For example for `):
print(`the differential equation satisfied by the Legendre polynomials. Try:`):
print(`For example AZpapc(1/sqrt(1-2*x*t+t^2)/t^(n+1),t,x); `):
 
fi:
 
 
 
if nops([args])=1 and op(1,[args])=`AZpapd` then
print(`AZpapd(INTEGRAND,x,n) inputs a hypergeometric integrand`):
print(`in the continuous variable x and the discrete variable n`):
print(`i.e. i.e. A(x,n+1)/A(x,n) and A'(x,n)/A(x,n) are rational functions`):
print(`of (x,n)),`):
print(`and outputs a paper with a proof of the linear recurrence equation`):
print(`that the definite integral w.r.t. to y (which is a function of n)`):
print(`assuming that the integrand (or rather it times the certificate`):
print(`vanishes at the endpoints, or it is a  contour integrals`):
print(`satisfies. It assumes the following: A(x,n) is not a product of`):
print(`of a function of n and a function of x`):
print(`It uses the method of`):
print(`Gert Almkvist and Doron Zeilberger, "The method of differentiating`):
print(`under the integral sign", J. Symbolic Computation 10(1990), 571-591`):
print(`It could be used to establish the recurrences of the `):
print(`classical orthogonal polynomials, when they are defined in terms`):
print(`or their generating funtion`):
print(`For example  for `):
print(`the recurrence, and proof, satisfied by the Legendre polynomials, try: `):
print(`AZpapd(1/sqrt(1-2*x*t+t^2)/t^(n+1),t,n) ; `):
 
fi:
 
 
if nops([args])=1 and op(1,[args])=`AZd` then
print(`AZd(A,x,n,N) finds a recurrence of order ORDER<=8`):
print(`phrased in terms of n, and the shift in n, N`):
print(`for the integral of A(x,n) with respect to x, whenever A(x,n) is`):
print(`hypergeometric in (x,n),i.e. A(x,n+1)/A(x,n) and A'(x,n)/A(x,n) are`):
print(`rational funtions of x and n. It follows the algorithn of`):
print(`Gert Almkvist and Doron Zeilberger, "The method of differentiating`):
print(`under the integral sign", J. Symbolic Computation 10(1990), 571-591`):
print(`A should not be a product of a function of x and a function of n.`):
print(``):
print(`AZd(A,x,n,N) returns the expression seq. ope(n,N),cert(x,n)`):
print(`satisfying ope(n,N)A(x,n)=diff(cert(x,n)*A(x,n),x).`):
print(`If no recurrence is found, it returns 0.`):
print(``):
print(`A verbose version is AZpapd(A,x,n), type ezra(AZpapd) for details.`):
print(``):
print(`For example for `):
print(`the recurrence equation and proof certificate, for the Legendre polct:ynomials. Try: `):
print(`AZd(1/sqrt(1-2*x*t+t^2)/t^(n+1),t,n,N) ; `):
fi:

if nops([args])=1 and op(1,[args])=`AZdI` then
print(`AZdI(A,x,n,N,a,b) finds an INHOMOGENEOUS recurrence of order ORDER<=8`):
print(`phrased in terms of n, and the shift in n, N`):
print(`for the integral of A(x,n) with respect to x, from x=a to x=b, whenever A(x,n) is`):
print(`hypergeometric in (x,n),i.e. A(x,n+1)/A(x,n) and A'(x,n)/A(x,n) are`):
print(`rational funtions of x and n. It follows the algorithn of`):
print(`Gert Almkvist and Doron Zeilberger, "The method of differentiating`):
print(`under the integral sign", J. Symbolic Computation 10(1990), 571-591`):
print(`A should not be a product of a function of x and a function of n.`):
print(``):
print(`AZd(A,x,n,N) returns the pair [ope(n,N),rhs(n)],cert(x,n)]`):
print(`satisfying ope(n,N)A(x,n)=diff(cert(x,n)*A(x,n),x).`):
print(`and rhs(n) is the right hand side of the inhomogeneous recurrence`):
print(`If no recurrence is found, it returns 0.`):
print(``):
print(``):
print(`For example try: `):
print(`AZdI(x^n/(1+x^2),x,n,N,-1,1) ; `):
fi:
 
 
if nops([args])=1 and op(1,[args])=`AZc` then
print(`AZc(A,y,x,D) tries to finds a linear diff.eq. of order <=10,`):
print(` phrased in terms of x, and diff.w.r.t x, D`):
print(`for the integrale of A(x,y) with respect to x, whenever A(x,y) is`):
print(`hypergeometric in (x,y),i.e.  A_x(x,y)/A(x,y) and A_y(x,y)/A(x,y) are`):
print(`rational funtions of x and y. It follows the algorithn of`):
print(`Gert Almkvist and Doron Zeilberger, "The method of differentiating`):
print(`under the integral sign", J. Symbolic Computation 10(1990), 571-591`):
print(`AZc(A,y,x,D) returns the expression seq. ope(x,D),cert(x,n)`):
print(`satisfying ope(x,D)A(x,y)=diff(cert(x,y)*A(x,y),y).`):
print(`If no linear diff. eq. of order<=8 is found, it returns 0`):
print(`see AZpapc for a verbose vsersion`):
print(`For example, for `): 
print(`the diff.eq., and proof certificate for the Legendre polynomials.`):
print(` AZc(1/sqrt(1-2*x*t+t^2)/t^(n+1),t,x,D); `):
fi:

if nops([args])=1 and op(1,[args])=`AZcI` then
print(`AZcI(A,y,x,D,a,b) tries to finds an inhomogeneous linear diff.eq. of order <=10,`):
print(` phrased in terms of x, and diff.w.r.t x, D`):
print(`for the integrale of A(x,y) with respect to x from x=a to x=b, whenever A(x,y) is`):
print(`hypergeometric in (x,y),i.e.  A_x(x,y)/A(x,y) and A_y(x,y)/A(x,y) are`):
print(`rational funtions of x and y. It follows the algorithn of`):
print(`Gert Almkvist and Doron Zeilberger, "The method of differentiating`):
print(`under the integral sign", J. Symbolic Computation 10(1990), 571-591`):
print(`AZc(A,y,x,D) returns the expression seq. ope(x,D),cert(x,n)`):
print(`satisfying ope(x,D)A(x,y)=diff(cert(x,y)*A(x,y),y).`):
print(`If no linear diff. eq. of order<=10 is found, it returns 0`):
print(`For example Try:`):
print(` AZcI(1/(1-x*t-(x+1)*t^2),x,t,D,0,1); `):
print(` AZcI(x*(2-x)/(1-x*t-(x+1)*t^2),x,t,D,0,2); `):
fi:
 
 
 if nops([args])=1 and op(1,[args])=`ct` then
 
 print(` ct(SUMMAND,ORDER,k,n,N)`): 
print(`This is a Maple inplementation of the algorithm described in Ch. 6`):
print(`of the book A=B, first proposed in : D. Zeilberger, "The method of`):
print(`of creative telescoping", J.Symbolic Computation 11(1991) 195-204`):
print(``):
print(`finds a recurrence for SUMMAND in the parameters k and n,   `):
print(` of order ORDER, with N is the chosen symbol for the shift in n.`):
 print(``):
 print(`SUMMAND should be a product of factorials and/or binomial coeffs`):
 print(`and/or rising factorials, where (a)_k is denoted by rf(a,k)`):
 print(`and/or powers of k and n, and, optionally, a polynomial factor`):
print(`The output consists of an operator ope(N,n) and a certificate R(n,k)`):
print(`with the properties that if we define G(n,k):=R(n,k)*SUMMAND then`):
print(`ope(N,n)SUMMAND(n,k)=G(n,k+1)-G(n,k)`):
print(`which is a routinely verifiable identity.`):
 print(`For example "ct(binomial(n,k),1,k,n,N);" would yield the output`):
print(`  N-2, k/(k-n-1) `):
 fi:

if nops([args])=1 and op(1,[args])=TerryTao then
print(`TerryTao(P,Q,x,z,k,L,K): inputs hyper-exponential functions P and Q of the variable x, `):
print(`a continuous variable z (for the certificate),`):
print(`a discrete variable`):
print(`k, and an affine-linear expression L in k of the form a0+b0*k for some non-negative integers a0 and b0`):
print(`and a symbol,K, for the shift operator in k.`):
print(`Outputs a recurrence, and a certificate (using the Almkvist-Zeilberger algorithm) for the`):
print(`sequence , in k, of the functions (of x)`):
print(`D_x^L(Q(x)*P(x)^k). For example, for the problem in Terry Tao's blog, May 30, 2015`):
print(` A differentiation identity, type `):
print(` TerryTao(1+x^2,(1+x^2)^(-1/2),x,z,k,2*k,K); `):
fi:
 
if nops([args])=1 and op(1,[args])=`zeil` then
print(`The syntax is:`):
 print(` zeil(SUMMAND,k,n,N) or zeil(SUMMAND,k,n,N,MAXORDER) or  `):
 print(` zeil(SUMMAND,k,n,N,MAXORDER,parameter_list) `):
 print(` finds a linear recurrence equation for SUMMAND, with`):
 print(` polynomial coefficients`):
 print(`of ORDER<=MAXORDER, where the default of MAXORDER is 6`):
 print(`in the parameter n, the shift operator in n being N`):
print(`of the form ope(N,n)SUMMAND=G(n,k+1)-G(n,k)`):
print(`where G(n,k):=R(n,k)*SUMMAND, and R(n,k) is the 2nd item of output.`):
print(`The output is ope(N,n),R(n,k) .`):
 print(`For example zeil(binomial(n,k),k,n,N) would yield`):
print(`  N-2, k/(k-n-1)  `);
print(` in which N-2 is the "ope" operator, and k/(k-n-1) is R(n,k) `);
 print(`SUMMAND should be a product of factorials and/or binomial coeffs`):
 print(`and/or rising factorials, where (a)_k is denoted by rf(a,k)`):
 print(`and/or powers in k and n, and, optionally, a polynomial factor.`):
 print(``):
 print(`The last optional parameter, is the list of other parameters,`):
 print(`if present. Giving them causes considerable speedup. For example`):
 print(` zeil(binomial(n,k)*binomial(a,k)*binomial(b,k),k,n,N,6,[a,b])`):
 
fi:
 
if nops([args])=1 and op(1,[args])=`zeilpap` then
print(` zeilpap(SUMMAND,k,n) or zeilpap(SUMMAND,k,n,NAME,REF)`):
print(`Just like zeil but writes a paper with the proof`):
print(`NAME and REF are optional name and reference`):
print(`Warning: It assumes that the definite summation w.r.t. k`):
print(`extends over all k where it is non-zero, and that it is zero`):
print(`for other k`):
print(`For non-natural summation limits, use zeillim`):
fi:
 
if nops([args])=1 and op(1,[args])=`zeillim` then
 print(` zeillim(SUMMAND,k,n,N,alpha,beta) `):
print(`Similar to zeil(SUMMAND,k,n,N) but outputs a recurrence for `):
print(` the sum of SUMMAND from k=alpha to k=n-beta .`):
print(`Outputs the recurrence operator, certificate and right hand side.`):
print(`Note carefully: The upper limit of the sum will be n-beta, not beta. `):
print(`For example, "zeillim(binomial(n,k),k,n,N,0,1);" gives output of `):
print(` N-2, k/(k-n-1),1 `):
print(` which means that SUM(n):=2^n-1 satisfies the recurrence `):
print(` (N-2)SUM(n)=1, as certified by R(n,k):=k/(k-n-1) `):
fi:
 
 
end:
 
#yafe just translates from operator notation to everyday notation
 
yafe:=proc(ope,N,n,SUM)
local gu,i:
gu:=0:
 
for i from 0 to degree(ope,N) do
 gu:=gu+coeff(ope,N,i)*subs(n=n+i,SUM):
od:
 
gu:
end:
 
 
 
yafec:=proc(ope,D,x,INTEGRAND)
local gu,i:
 
 gu:=coeff(ope,D,0)*INTEGRAND:
 
 
for i from 1 to degree(ope,D) do
 gu:=gu+coeff(ope,D,i)*diff(INTEGRAND,x$i):
od:
 
gu:
end:
 
  
simplify1:=proc(bitu,k,a)
local gu,gu1,i,khez,sp:
sp:=1:
gu:=bitu:
 
if type(gu,`*`) then
 
 for i from 1 to nops(gu) do
  gu1:=op(i,gu):
 
 if type(gu1,`^`) and type(op(2,gu1), integer) then
 
     khez:=op(2,gu1):
     gu1:=op(1,gu1):
     sp:=sp*simplify(subs(k=k+a,gu1)/gu1)^khez:
  
   else 
     sp:=sp*simplify(subs(k=k+a,gu1)/gu1):
 
   fi:
 
od:
 
elif type(gu,`^`) and type(op(2,gu), integer) then
   khez:=op(2,gu):
   gu1:=op(1,gu):
   sp:=sp*simplify(subs(k=k+a,gu1)/gu1)^khez:  
 
 
else
 
 sp:=simplify(subs(k=k+a,gu)/gu):
 
fi:
 
sp:
 
end:
 
 
rf:=proc(a,b):
 (a+b-1)!/(a-1)!:
end:
 
 
 
RootOf1:=proc(f,x)
local kv,kvi,i:
kv:=[solve(f=0,x)]:
kvi:={}:
for i from 1 to nops(kv) do
 
  if type(kv[i],integer) and kv[i]>0 then
   kvi:=kvi union {kv[i]}
  fi:
 
od:
 
RETURN(kvi):
 
end:
 
pashet:=proc(p,N)
local i,gu1,gu,p1:
p1:=normal(p):
gu1:=denom(p1):
p1:=numer(p1):
p1:=expand(p1):
gu:=0:
for i from 0 to degree(p,N) do
gu:=gu+factor(coeff(p1,N,i))*N^i:
od:
RETURN(gu/gu1):
end:
 
 
 
hafel:=proc(POL,SUMMAND,ope,n,N)
local i,FAC,degg,bi,rat,POL1,SUMMAND1,zub:
degg:=degree(ope,N):
FAC:=0:
 
for i from 0 to degg do
 bi:=coeff(ope,N,i):
 rat:=simplify1(SUMMAND,n,i):
 FAC:=FAC+bi*subs(n=n+i,POL)*rat:
 FAC:=normal(FAC):
od:
 
POL1:=numer(FAC):
zub:=denom(FAC):
SUMMAND1:=SUMMAND/zub:
RETURN(POL1,SUMMAND1,zub):
end:
 
ctold:=proc(SUMMAND,ORDER,k,n,N)
local gu,i,ope,POL1,SUMMAND1,yakhas,P,Q,R,j,res1,kv,hakhi,g,
A1, B1,  P1, SDZ, SDZ1,
degg, eq, fu, gugu, i1, ia1, k1, kvutsa, l1, l2, meka1, mekb1, mumu, 
va1, va2,ope1,denFAC,ope1a:
 
if nargs<>5 then
ERROR(`Syntax: ct(SUMMAND,ORDER,summation_variable,auxiliary_var, Shift_n)`):
fi:
 
 
ope:=0:
 
for i from 0 to ORDER do
 ope:=ope+bpc[i]*N^i:
od: 
gu:=hafel(1,convert(SUMMAND,factorial),ope,n,N):
POL1:=gu[1]:
SUMMAND1:=gu[2]:
denFAC:=gu[3]:
yakhas:=simplify1(SUMMAND1,k,-1):
yakhas:=normal(1/yakhas):
 
print(`yakhas is`,yakhas):
P1:=1:
Q:=numer(yakhas):
R:=denom(yakhas):
res1:=resultant(Q,subs(k=k+j,R),k):
print(`res1 is`,res1):
kv:=RootOf1(res1,j):
 
 
 print(`kv is`,kv):
while nops(kv) > 0 do
 hakhi:=max(op(kv)):
 g:=gcd(Q,subs(k=k+hakhi,R)):
 Q:=normal(Q/g):
 R:=normal(R/subs(k=k-hakhi,g)):
 P1:=P1*product(subs(k=k-i1,g),i1=0..hakhi-1):
 res1:=resultant(Q,subs(k=k+j,R),k):
 kv:=RootOf1(res1,j):
od:
 
P:=POL1*P1:
 
A1:=subs(k=k+1,Q)+R:
A1:=expand(A1):
B1:=subs(k=k+1,Q)-R:
B1:=expand(B1):
l1:=degree(A1,k):
if B1=0 then
l2:=-100:
else
l2:=degree(B1,k):
fi:
meka1:=coeff(A1,k,l1):
mekb1:=coeff(B1,k,l2):
if l1<=l2 
then
k1:=degree(P,k)-l2:
elif l2=l1-1 
then
 mumu:= (-2)*mekb1/meka1:
  if type(mumu,integer) 
  then
   k1:=max(mumu, degree(P,k)-l1+1):
  else
   k1:=degree(P,k)-l1+1:
  fi:
elif l2 < l1-1
then
 k1:= max( 0, degree(P,k)-l1+1 ):
fi:
fu:=0:
 
 
print(`k1 is`,k1):
if k1 < 0 then
RETURN(0):
fi:
if k1 >= 0 then
for ia1 from 0 to k1 do
fu:=fu+apc[ia1]*k^ia1:
od:
gugu:=subs(k=k+1,Q)*fu-R*subs(k=k-1,fu)-P:
gugu:=expand(gugu):
degg:=degree(gugu,k):
 
for ia1 from 0 to degg do
eq[ia1+1]:=coeff(gugu,k,ia1)=0:
od:
for ia1 from 0 to k1 do
va1[ia1+1]:=apc[ia1]:
od:
for ia1 from 0 to ORDER do
va2[ia1+1]:=bpc[ia1]:
od:
eq:=convert(eq,set):
va1:=convert(va1,set):
va2:=convert(va2,set):
va1:=va1 union va2:
va1:=solve1(eq,va1):
kvutsa:={va1}:
fu:=subs(va1,fu):
ope:=subs(va1,ope):
fi:
 
 
 
if ope=0 or kvutsa={} or fu=0 then
RETURN(0):
fi:
gu:={}:
for i1 from 0 to k1 do
gu:=gu union {apc[i1]=1}:
od:
for i1 from 0 to ORDER do
gu:=gu union {bpc[i1]=1}:
od:
fu:=subs(gu,fu):
ope:=subs(gu,ope):
 
ope:=pashet(ope,N):
ope1:=ope*denom(ope):
 
SDZ:=denom(ope)*subs(k=k+1,Q)*fu/P1/denFAC :
SDZ1:=subs(k=k-1,SDZ)*simplify1(convert(SUMMAND,factorial),k,-1):
SDZ1:=simplify(SDZ1):
 
ope1a:=0:
for i from 0 to degree(ope1,N) do
ope1a:=ope1a+sort(coeff(ope1,N,i)*N^i):
od:
 
RETURN(ope1a,SDZ1):
end:
 
 
 
 
ct:=proc(SUMMAND,ORDER,k,n,N)
local gu,i,ope,POL1,SUMMAND1,yakhas,P,Q,R,j,res1,kv,hakhi,g,
A1, B1,  P1, SDZ, SDZ1,
degg, eq, fu, gugu, i1, ia1, k1, kvutsa, l1, l2, meka1, mekb1, mumu, 
va1, va2,ope1,denFAC,ope1a:
 
if nargs<>5 then
ERROR(`Syntax: ct(SUMMAND,ORDER,summation_variable,auxiliary_var, Shift_n)`):
fi:
 
if tovu(convert(SUMMAND,factorial),k,n)=0 then
ERROR(`The summand can be separated into f(`,k,`)g(`,n,`)`): 
fi:
 
ope:=0:
 
for i from 0 to ORDER do
 ope:=ope+bpc[i]*N^i:
od:
 
gu:=hafel(1,convert(SUMMAND,factorial),ope,n,N):
POL1:=gu[1]:
SUMMAND1:=gu[2]:
denFAC:=gu[3]:
yakhas:=simplify1(SUMMAND1,k,-1):
yakhas:=normal(1/yakhas):
 
P1:=1:
Q:=numer(yakhas):
R:=denom(yakhas):
 
 
res1:=findgQR(Q,R,k,100):
 
while res1[2]<>0 do
 g:=res1[1]:
 hakhi:=res1[2]:
 Q:=normal(Q/g):
 R:=normal(R/subs(k=k-hakhi,g)):
 P1:=P1*product(subs(k=k-i1,g),i1=0..hakhi-1):
 res1:=findgQR(Q,R,k,100):
od:
 
P:=POL1*P1:
 
A1:=subs(k=k+1,Q)+R:
A1:=expand(A1):
B1:=subs(k=k+1,Q)-R:
B1:=expand(B1):
l1:=degree(A1,k):
if B1=0 then
l2:=-100:
else
l2:=degree(B1,k):
fi:
meka1:=coeff(A1,k,l1):
mekb1:=coeff(B1,k,l2):
if l1<=l2 
then
k1:=degree(P,k)-l2:
elif l2=l1-1 
then
 mumu:= (-2)*mekb1/meka1:
  if type(mumu,integer) 
  then
   k1:=max(mumu, degree(P,k)-l1+1):
  else
   k1:=degree(P,k)-l1+1:
  fi:
elif l2 < l1-1
then
 k1:= max( 0, degree(P,k)-l1+1 ):
fi:
fu:=0:
 
 
if k1 < 0 then
RETURN(0):
fi:
if k1 >= 0 then
for ia1 from 0 to k1 do
fu:=fu+apc[ia1]*k^ia1:
od:
gugu:=subs(k=k+1,Q)*fu-R*subs(k=k-1,fu)-P:
gugu:=expand(gugu):
degg:=degree(gugu,k):
 
for ia1 from 0 to degg do
eq[ia1+1]:=coeff(gugu,k,ia1)=0:
od:
for ia1 from 0 to k1 do
va1[ia1+1]:=apc[ia1]:
od:
for ia1 from 0 to ORDER do
va2[ia1+1]:=bpc[ia1]:
od:
eq:=convert(eq,set):
va1:=convert(va1,set):
va2:=convert(va2,set):
va1:=va1 union va2:
va1:=solve1(eq,va1):
kvutsa:={va1}:
fu:=subs(va1,fu):
ope:=subs(va1,ope):
fi:
 
 
 
if ope=0 or kvutsa={} or fu=0 then
RETURN(0):
fi:
gu:={}:
for i1 from 0 to k1 do
gu:=gu union {apc[i1]=1}:
od:
for i1 from 0 to ORDER do
gu:=gu union {bpc[i1]=1}:
od:
fu:=subs(gu,fu):
ope:=subs(gu,ope):
 
ope:=pashet(ope,N):
ope1:=ope*denom(ope):
 
SDZ:=denom(ope)*subs(k=k+1,Q)*fu/P1/denFAC :
SDZ1:=subs(k=k-1,SDZ)*simplify1(convert(SUMMAND,factorial),k,-1):
SDZ1:=simplify(SDZ1):
 
ope1a:=0:
for i from 0 to degree(ope1,N) do
ope1a:=ope1a+sort(coeff(ope1,N,i)*N^i):
od:
 
RETURN(ope1a,SDZ1):
end:
 
 
 
cttry:=proc(SUMMAND,ORDER,k,n,resh,N)
local gu,i,ope,POL1,SUMMAND1,yakhas,P,Q,R,j,res1,kv,hakhi,g,
A1, B1,  P1, 
degg, eq, fu, gugu, i1, ia1, k1, kvutsa, l1, l2, meka1, mekb1, mumu, 
va1, va2,eqg:
 
if nargs<>6 then
ERROR(`The syntax is cttry(term,ORDER,sum_variable,aux_var,para_list,Shift)`):
fi:
 
if tovu(convert(SUMMAND,factorial),k,n)=0 then
ERROR(`The summand can be separated into f(`,k,`)g(`,n,`)`): 
fi:
 
ope:=0:
 
for i from 0 to ORDER do
 ope:=ope+bpc[i]*N^i:
od:
 
gu:=hafel(1,convert(SUMMAND,factorial),ope,n,N):
POL1:=gu[1]:
SUMMAND1:=gu[2]:
yakhas:=simplify1(SUMMAND1,k,-1):
yakhas:=normal(1/yakhas):
 
P1:=1:
Q:=numer(yakhas):
R:=denom(yakhas):
 
res1:=findgQR(Q,R,k,100):
 
while res1[2]<>0 do
 g:=res1[1]:
 hakhi:=res1[2]:
 Q:=normal(Q/g):
 R:=normal(R/subs(k=k-hakhi,g)):
 P1:=P1*product(subs(k=k-i1,g),i1=0..hakhi-1):
 res1:=findgQR(Q,R,k,100):
od:
 
P:=POL1*P1:
 

A1:=subs(k=k+1,Q)+R:
A1:=expand(A1):
B1:=subs(k=k+1,Q)-R:
B1:=expand(B1):
l1:=degree(A1,k):
if B1=0 then
l2:=-100:
else
l2:=degree(B1,k):
fi:
meka1:=coeff(A1,k,l1):
mekb1:=coeff(B1,k,l2):
if l1<=l2 
then
k1:=degree(P,k)-l2:
 if k1=FAIL then
   RETURN(0):
 fi:

elif l2=l1-1 
then
 mumu:= (-2)*mekb1/meka1:
  if type(mumu,integer) 
  then
   k1:=max(mumu, degree(P,k)-l1+1):
  else
   k1:=degree(P,k)-l1+1:
  fi:
elif l2 < l1-1
then

 k1:= max( 0, degree(P,k)-l1+1 ):

fi:
fu:=0:
 

 
if k1 < 0 then
RETURN(0):
fi:
if k1 >= 0 then
for ia1 from 0 to k1 do
fu:=fu+apc[ia1]*k^ia1:
od:
gugu:=subs(k=k+1,Q)*fu-R*subs(k=k-1,fu)-P:
gugu:=expand(gugu):
degg:=degree(gugu,k):
for ia1 from 0 to degg do
eq[ia1+1]:=coeff(gugu,k,ia1)=0:
od:
for ia1 from 0 to k1 do
va1[ia1+1]:=apc[ia1]:
od:
for ia1 from 0 to ORDER do
va2[ia1+1]:=bpc[ia1]:
od:
eq:=convert(eq,set):
va1:=convert(va1,set):
va2:=convert(va2,set):
va1:=va1 union va2:
eqg:=subs(n=1/2,eq):
 
for i from 1 to nops(resh) do
 eqg:=subs(op(i,resh)=i^2+3,eqg):
od:
 
va1:=solve1(eq,va1):
kvutsa:={va1}:
fu:=subs(va1,fu):
ope:=subs(va1,ope):
fi:
 
 
 
if ope=0 or kvutsa={} or fu=0 then
RETURN(0):
 else
  RETURN(1):
fi:
 
end:
 
 
paper:=proc(SUMMAND,k,n,N,ope1,SDZ1,NAME,REF)
local SHEM,IDENTITY,RECURRENCE:
 
if degree(ope1,N)=1 then
SHEM:=IDENTITY:
else
SHEM:=RECURRENCE:
fi:
print(``):
print(`A PROOF OF THE`,NAME,SHEM):
print(``):
print(`By Shalosh B. Ekhad,Rutgers University, c/o zeilberg@math.rutgers.edu`):
print(``):
print(`I will give a short proof of the following result(`,REF,`).`):
print(``):
if degree(ope1,N)=1 then
print(`(Note that since the recurrence below is first order, this`):
print(`means that the sum`, SUM(n), `has closed form,and it is`):
print(`easily seen to be equivalent.)`):
print(``):
fi:
print(`Theorem:Let`, F(n,k), `be given by`):
print(``):
print(SUMMAND):
print(``):
print(`and let`, SUM(n),`be the sum of`, F(n,k),` with`):
print(`respect to`, k):
print(``):
print(SUM(n),` satisfies the following linear recurrence equation`):
print(``):
print(yafe(ope1,N,n,SUM(n))):
print(`=0.`):
print(``):
print(`PROOF: We cleverly construct`, G(n,k),`:=`):
print(``):
print(SDZ1*SUMMAND):
print(`with the motive that`):
print(``):
print(yafe(ope1,N,n,F(n,k)) = G(n,k+1)-G(n,k), `(Check!)`):
print(``):
print(`and the theorem follows upon summing with respect to`, k,`. QED.`):
end:
 
 
 
paper3:=proc(SUMMAND,k,n,N,ope1,SDZ1):
 
print(``):
print(`A PROOF OF A RECURRENCE`):
print(``):
print(`By Shalosh B. Ekhad, Rutgers University `):
print(``):
 
print(`Theorem: Let`, F(n,k), `be given by`):
print(``):
print(SUMMAND):
print(``):
print(`and let`, SUM(n),`be the sum of`, F(n,k),` with`):
print(`respect to`, k):
print(``):
print(SUM(n),` satisfies the following linear recurrence equation`):
print(``):
print(yafe(ope1,N,n,SUM(n))):
print(`=0.`):
print(``):
print(`PROOF: We cleverly construct`, G(n,k),`:=`):
print(``):
print(SDZ1*SUMMAND):
print(`with the motive that`):
print(``):
print(yafe(ope1,N,n,F(n,k))=G(n,k+1) - G(n,k), `(Check!)`):
print(``):
print(`and the theorem follows upon summing with respect to`, k,`. QED.`):
end:
 
 
paperc:=proc(INTEGRAND,y,x,D,ope1,SDZ1):
print(``):
print(``):
print(`A PROOF OF A DIFFERENTIAL EQUATION SATISFIED BY AN INTEGRAL`):
print(``):
print(`By Shalosh B. Ekhad, Rutgers University, c/o zeilberg@math.rutgers.edu`):
print(``):
print(`I will give a short proof of the following result.`):
print(``):
 
print(`Theorem:Let`, F(x,y), `be given by`):
print(``):
print(INTEGRAND):
print(``):
print(`and let`, INTEGRAL(x),`be the integral of`, F(x,y),` with`):
print(`respect to`, y):
print(``):
print(INTEGRAL(x),` satisfies the following linear differential equation`):
print(``):
print(yafec(ope1,D,x,INTEGRAL(x))):
print(`=0.`):
print(``):
print(`PROOF: We cleverly construct`, G(x,y),`:=`):
print(``):
print(SDZ1*INTEGRAND):
print(`with the motive that`):
print(``):
print(yafec(ope1,D,x,F(x,y)) = diff(G(x,y),y),  `(Check!)`):
print(``):
print(`and the theorem follows upon integrating with respect to`, y):
end:
 
paperd:=proc(INTEGRAND,x,n,N,ope1,SDZ1):
print(``):
print(`A PROOF OF A LINEAR RECURRENCE SATISFIED BY AN INTEGRAL`):
print(``):
print(`By Shalosh B. Ekhad, Rutgers University, c/o zeilberg@math.rutgers.edu`):
print(``):
print(`I will give a short proof of the following result.`):
print(``):
 
print(`Theorem:Let`, F(n,x), `be given by`):
print(``):
print(INTEGRAND):
print(``):
print(`and let`, INTEGRAL(n),`be the integral of`, F(n,x),` with`):
print(`respect to`, x):
print(``):
print(INTEGRAL(n),` satisfies the following linear recurrence equation`):
print(``):
print(yafe(ope1,N,n,INTEGRAL(n))):
print(`=0.`):
print(``):
print(`PROOF: We cleverly construct`, G(n,x),`:=`):
print(``):
print(SDZ1*INTEGRAND):
print(`with the motive that`):
print(``):
print(yafe(ope1,N,n,F(n,x)) = diff(G(n,x),x)):
print(``):
print(`and the theorem follows upon integrating with respect to`, x,`. QED.`):
end:
 
 
paperc:=proc(INTEGRAND,y,x,D,ope1,SDZ1):
print(`A PROOF OF A DIFFERENTIAL EQUATION SATISFIED BY AN INTEGRAL`):
print(``):
print(`By Shalosh B. Ekhad, Rutgers University`):
print(``):
print(`I will give a short proof of the following result.`):
print(``):
 
print(`Theorem:Let`, F(x,y), `be given by`):
print(``):
print(INTEGRAND):
print(``):
print(`and let`, INTEGRAL(x),`be the integral of`, F(x,y),` with`):
print(`respect to`, y):
print(``):
print(INTEGRAL(x),` satisfies the following linear differential equation`):
print(``):
print(yafec(ope1,D,x,INTEGRAL(x))):
print(`=0.`):
print(``):
print(`PROOF: We cleverly construct`, G(x,y),`:=`):
print(``):
print(SDZ1*INTEGRAND):
print(`with the motive that`):
print(``):
print(yafec(ope1,D,x,F(x,y)) = diff(G(x,y),y), `(Check!)`):
print(``):
print(`and the theorem follows upon integrating with respect to`, y,`. QED.`):
end:
 
 
 
zeil4:=proc(SUMMAND,k,n,N)
local ORDER,MAXORDER,gu,gu1,resh:
 
 
 MAXORDER:=6:
 
  resh:=[]:
 
 
if simplify1(SUMMAND,n,1)=1 then
ERROR(`Summand does not depend on`,n,` hence the sum is identically constant`):
fi:
for ORDER from 0 to MAXORDER do
 gu1:=cttry(SUMMAND,ORDER,k,n,resh,N):
 if gu1=1 then
  gu:=ct(SUMMAND,ORDER,k,n,N):
 
   if gu<>0 then
     RETURN(gu):
   fi:
 fi:
od:
 
FAIL:
 
end:
 
 
 
 
zeil5:=proc(SUMMAND,k,n,N,MAXORDER)
local ORDER,gu,gu1,resh:
 
resh:=[]:
 
if simplify1(SUMMAND,n,1)=1 then
ERROR(`Summand does not depend on`,n,` hence the sum is identically constant`):
fi:
for ORDER from 0 to MAXORDER do
 gu1:=cttry(SUMMAND,ORDER,k,n,resh,N):
 if gu1=1 then
  gu:=ct(SUMMAND,ORDER,k,n,N):
 
   if gu<>0 then
     RETURN(gu):
   fi:
 fi:
od:
 
FAIL:
 
end:
 
 
 
 
zeil6:=proc(SUMMAND,k,n,N,MAXORDER,resh)
local ORDER,gu,gu1:
 
 
if simplify1(SUMMAND,n,1)=1 then
ERROR(`Summand does not depend on`,n,` hence the sum is identically constant`):
fi:
for ORDER from 0 to MAXORDER do
 gu1:=cttry(SUMMAND,ORDER,k,n,resh,N):
 if gu1=1 then
  gu:=ct(SUMMAND,ORDER,k,n,N):
 
   if gu<>0 then
     RETURN(gu):
   fi:
 fi:
od:
 
FAIL:
 
end:
 
 
zeil:=proc():
 
if nargs=4 then
 zeil4(args):
elif nargs=5 then
 zeil5(args):
elif nargs=6 then
 zeil6(args):
else
  print(`zeil(SUMMAND,k,n,N) or zeil(SUMMAND,k,n,N,MAXORDER) or`):
  ERROR(`zeil(SUMMAND,k,n,N,MAXORDER,param_list)`):
fi:
 
 
 
end:
 
 
 
 
 
zeillim:=proc(SUMMAND,k,n,N,alpha,beta)
local ope,CERT,lu,k1,i,gu,lu1,lu2:
 
gu:=Zeillim(SUMMAND,k,n,N,alpha+1,beta+1):
ope:=gu[1]:
CERT:=gu[2]:
lu:=gu[3]: 
lu1:=subs(k=alpha,SUMMAND)+subs(k=n-beta,SUMMAND):
lu1:=simplify(lu1):
lu1:=normal(lu1):
lu2:=0:
for i from 0 to degree(ope,N) do
   lu2:=lu2+coeff(ope,N,i)*simplify(subs(n=n+i,lu1)):
 od:
 lu2:=normal(lu2):
ope,CERT,normal(expand(normal(simplify(expand(normal(lu+lu2)))))):
 
end:
 
Zeillim:=proc(SUMMAND,k,n,N,alpha,beta)
local ope,CERT,lu,k1,i,gu:
 
gu:=zeil(SUMMAND,k,n,N):
ope:=gu[1]:
CERT:=gu[2]:
 
lu:=simplify(subs(k=n-beta+1,CERT)*subs(k=n-beta+1,SUMMAND))
-simplify(subs(k=alpha,CERT)*subs(k=alpha,SUMMAND)):
 
for i from 0 to degree(ope,N) do
 for k1 from 1 to i do
   lu:=lu+coeff(ope,N,i)*simplify(subs({n=n+i,k=n-beta+k1},SUMMAND)):
 od:
od:
 
gu,normal(expand(lu)):
 
end:
 
 
 
zeilpap3:=proc(SUMMAND,k,n)
local  SDZ1, gu, ope1,N:
 
gu:=zeil4(SUMMAND,k,n,N):
ope1:=gu[1]:
SDZ1:=gu[2]:
paper3(SUMMAND,k,n,N,ope1,SDZ1):
end:
 
 
 
 
zeilpap5:=proc(SUMMAND,k,n,NAME,REF)
local SDZ1, gu, ope1,N:
 
gu:=zeil4(SUMMAND,k,n,N):
ope1:=gu[1]:
SDZ1:=gu[2]:
paper(SUMMAND,k,n,N,ope1,SDZ1,NAME,REF):
end:
 
 zeilpap:=proc()
 
if nargs=5 then
zeilpap5(args):
elif nargs=3 then
zeilpap3(args):
else
 ERROR(`zeilpap(SUMMAND,k,n,NAME,REF) or  zeilpap(SUMMAND,k,n)`):
fi:
end:
 
 
 
 
AZpapc:=proc(INTEGRAND,y,x)
local D,SDZ1,gu,ope1:
gu:=AZc(INTEGRAND,y,x,D):
ope1:=gu[1]:
SDZ1:=gu[2]:
paperc(INTEGRAND,y,x,D,ope1,SDZ1):
end:
 
 
 
 
AZpapd:=proc(INTEGRAND,x,n)
local D,SDZ1, gu, ope1:
gu:=AZd(INTEGRAND,x,n,D):
ope1:=gu[1]:
SDZ1:=gu[2]:
paperd(INTEGRAND,x,n,D,ope1,SDZ1):
end:
 
 
 
 
 
goremd:=proc(N,ORDER)
local i, gu:
gu:=0:
for i from 0 to ORDER do
gu:=gu+(bpc[i])*N^i:
od:
RETURN(gu):
end:
 
goremc:=proc(D,ORDER)
local i,gu:
gu:=0:
for i from 0 to ORDER do
gu:=gu+(bpc[i])*D^i:
od:
RETURN(gu):
end:
 
gzor:=proc(f,x,n)
local i,gu:
gu:=f:
for i from 1 to n do
gu:=diff(gu,x):
od:
RETURN(gu):
end:
 
gzor1:=proc(a,D,x)
local i,gu:
gu:=0:
for i from 0 to degree(a,D) do
gu:=gu+diff(coeff(a,D,i),x)*D^i+coeff(a,D,i)*D^(i+1):
od:
end:
 
pashetd:=proc(p,N)
local gu,p1,mekh,i:
p1:=normal(p):
mekh:=denom(p1):
p1:=numer(p1):
p1:=expand(p1):
gu:=0:
for i from 0 to degree(p1,N) do
gu:=gu+factor(coeff(p1,N,i))*N^i:
od:
RETURN(gu,mekh):
end:
 
pashetc:=proc(p,D)
local gu,p1,i,mekh:
p1:=normal(p):
mekh:=denom(p1):
p1:=numer(p1):
p1:=expand(p1):
gu:=0:
for i from 0 to degree(p1,D) do
gu:=gu+factor(coeff(p1,D,i))*D^i:
od:
RETURN(gu,mekh):
end:
 
 
AZd:= proc(A,y,n,N)
local ORDER,gu,KAMA:
KAMA:=8:
 
for ORDER from 0 to KAMA do
gu:=duisd(A,ORDER,y,n,N):
if gu<>0 then
 RETURN(gu):
fi:
od:
0:
end:
 
 
AZc:=proc(A,y,x,D)
local ORDER,gu,KAMA:
KAMA:=10:
 
for ORDER from 1 to KAMA do
gu:=duisc(A,ORDER,y,x,D):
if gu<>0 and gu[1]<>0 then
 RETURN(gu):
fi:
od:
0:
end:
 
 
duisd:= proc(A,ORDER,y,n,N)
local gorem, conj, yakhas,lu1a,LU1,P,Q,R,S,j1,g,Q1,Q2,l1,l2,mumu,
mekb1,fu,meka1,k1,gugu,eq,ia1,va1,va2,degg,shad,KAMA,i1:
KAMA:=10:
gorem:=goremd(N,ORDER):
 
 
conj:=gorem:
yakhas:=0:
 
for i1 from 0 to degree(conj,N) do
 
yakhas:=yakhas+coeff(conj,N,i1)*simplify(subs(n=n+i1,A)/A):
od:
 
lu1a:=yakhas:
LU1:=numer(yakhas):
yakhas:=1/denom(yakhas):
yakhas:=normal(diff(yakhas,y)/yakhas+diff(A,y)/A):
P:=LU1:
Q:=numer(yakhas):
R:=denom(yakhas):
j1:=0:
while j1 <= KAMA do
g:=gcd(R,Q-j1*diff(R,y)):
if g <> 1 then
Q2:=(Q-j1*diff(R,y))/g:
R:=normal(R/g):
Q:=normal(Q2+j1*diff(R,y)):
P:=P*g^j1:
j1:=-1:
fi:
j1:=j1+1:
od:
P:=expand(P):
R:=expand(R):
Q:=expand(Q):
Q1:=Q+diff(R,y):
Q1:=expand(Q1):
l1:=degree(R,y):
l2:=degree(Q1,y):
meka1:=coeff(R,y,l1):
mekb1:=coeff(Q1,y,l2):
if l1-1 <>l2 
then
k1:=degree(P,y)-max(l1-1,l2):
else
 mumu:= -mekb1/meka1:
  if type(mumu,integer) and mumu > 0 
  then
   k1:=max(mumu, degree(P,y)-l2):
  else
   k1:=degree(P,y)-l2:
  fi:
fi:
fu:=0:
if k1 < 0 then
RETURN(0):
fi:
for ia1 from 0 to k1 do
fu:=fu+apc[ia1]*y^ia1:
od:
gugu:=Q1*fu+R*diff(fu,y)-P:
gugu:=expand(gugu):
degg:=degree(gugu,y):
for ia1 from 0 to degg do
eq[ia1+1]:=coeff(gugu,y,ia1)=0:
od:
for ia1 from 0 to k1 do
va1[ia1+1]:=apc[ia1]:
od:
for ia1 from 0 to ORDER do
va2[ia1+1]:=bpc[ia1]:
od:
eq:=convert(eq,set):
va1:=convert(va1,set):
va2:=convert(va2,set):
va1:=va1 union va2 :
va1:=solve1(eq,va1):
fu:=subs(va1,fu):
gorem:=subs(va1,gorem):
if fu=0 and gorem=0 then
 RETURN(0):
fi:
for ia1 from 0 to k1 do
gorem:=subs(apc[ia1]=1,gorem):
od:
for ia1 from 0 to ORDER do
gorem:=subs(bpc[ia1]=1,gorem):
od:
fu:=normal(fu):
 
 
shad:=pashetd(gorem,N):
S:=lu1a*R*fu*shad[2]/P:
S:=subs(va1,S):
S:=normal(S):
S:=factor(S):
for ia1 from 0 to k1 do
S:=subs(apc[ia1]=1,S):
od:
for ia1 from 0 to ORDER do
S:=subs(bpc[ia1]=1,S):
od:
 
RETURN(shad[1],S):
end:
 
 
duisc:= proc(A,ORDER,y,x,D)
local KAMA,gorem,conj, yakhas,lu1a,LU1,P,Q,R,S,j1,g,Q1,Q2,l1,l2,mumu,
mekb1,fu,meka1,k1,gugu,eq,ia1,va1,va2,degg,i,shad:
 
KAMA:=10:
gorem:=goremc(D,ORDER):
 
conj:=gorem:
yakhas:=0:
 
for i from 0 to degree(conj,D) do
yakhas:=yakhas+normal(coeff(conj,D,i)*gzor(A,x,i)/A):
yakhas:=normal(yakhas):
od:
 
lu1a:=yakhas:
LU1:=numer(yakhas):
yakhas:=1/denom(yakhas):
yakhas:=normal(diff(yakhas,y)/yakhas+diff(A,y)/A):
P:=LU1:
Q:=numer(yakhas):
R:=denom(yakhas):
j1:=0:
while j1 <= KAMA do
g:=gcd(R,Q-j1*diff(R,y)):
if g <> 1 then
Q2:=(Q-j1*diff(R,y))/g:
R:=normal(R/g):
Q:=normal(Q2+j1*diff(R,y)):
P:=P*g^j1:
j1:=-1:
fi:
j1:=j1+1:
od:
P:=expand(P):
R:=expand(R):
Q:=expand(Q):
Q1:=Q+diff(R,y):
Q1:=expand(Q1):
l1:=degree(R,y):
l2:=degree(Q1,y):
meka1:=coeff(R,y,l1):
mekb1:=coeff(Q1,y,l2):
if l1-1 <>l2 
then
k1:=degree(P,y)-max(l1-1,l2):
else
 mumu:= -mekb1/meka1:
  if type(mumu,integer) and mumu > 0 
  then
   k1:=max(mumu, degree(P,y)-l2):
  else
   k1:=degree(P,y)-l2:
  fi:
fi:
fu:=0:
if k1 < 0 then
RETURN(0):
fi:
for ia1 from 0 to k1 do
fu:=fu+apc[ia1]*y^ia1:
od:
gugu:=Q1*fu+R*diff(fu,y)-P:
gugu:=expand(gugu):
degg:=degree(gugu,y):
for ia1 from 0 to degg do
eq[ia1+1]:=coeff(gugu,y,ia1)=0:
od:
for ia1 from 0 to k1 do
va1[ia1+1]:=apc[ia1]:
od:
for ia1 from 0 to ORDER do
va2[ia1+1]:=bpc[ia1]:
od:
eq:=convert(eq,set):
va1:=convert(va1,set):
va2:=convert(va2,set):
va1:=va1 union va2 :
va1:=solve1(eq,va1):
fu:=subs(va1,fu):
gorem:=subs(va1,gorem):
if fu=0 and gorem=0 then
 RETURN(0):
fi:
for ia1 from 0 to k1 do
gorem:=subs(apc[ia1]=1,gorem):
od:
for ia1 from 0 to ORDER do
gorem:=subs(bpc[ia1]=1,gorem):
od:
fu:=normal(fu):
shad:=pashetc(gorem,D):
S:=lu1a*R*fu*shad[2]/P:
S:=subs(va1,S):
S:=normal(S):
S:=factor(S):
for ia1 from 0 to k1 do
S:=subs(apc[ia1]=1,S):
od:
for ia1 from 0 to ORDER do
S:=subs(bpc[ia1]=1,S):
od:
shad[1],S:
end:
 
 
 
bdokcertc:=proc(A,y,x,D,ope,S)
local gu,i:
gu:=0:
 
for i from 0 to degree(ope,D) do
gu:=gu+coeff(ope,D,i)*simplify(gzor(A,x,i)/A):
gu:=normal(gu):
od:
 
gu:=gu/simplify(diff(S*A,y)/A):
normal(gu);
end:
 
 
bdokcertd:=proc(A,y,n,N,ope,S)
local gu,i:
gu:=0:
 
for i from 0 to degree(ope,N) do
gu:=gu+coeff(ope,N,i)*simplify( subs(n=n+i,A)/A):
gu:=normal(gu):
od:
 
gu:=gu/simplify(diff(S*A,y)/A):
normal(gu);
end:
 
 
 
bdokcto:=proc(SUMMAND1,ORDER,k,n,N)
local mu,gu,i,G1,ope,lu,SUMMAND:
 
SUMMAND:=convert(SUMMAND1,factorial):
mu:=ct(SUMMAND,ORDER,k,n,N):
 
if mu=0 then
 RETURN(0):
fi:
 
ope:=mu[1]:
G1:=mu[2]*SUMMAND:
gu:=0:
 
for i from 0 to degree(ope,N) do
gu:=gu+coeff(ope,N,i)*simplify( subs(n=n+i,SUMMAND)/SUMMAND):
gu:=normal(gu):
od:
 
lu:=simplify(subs(k=k+1,G1)/SUMMAND)-mu[2]:
lu:=normal(lu):
normal(gu/lu);
end:
 
 
 
 
bdokct:=proc(SUMMAND1,ORDER,k,n,N)
local mu,gu,i,G1,ope,lu,SUMMAND:
 
SUMMAND:=convert(SUMMAND1,factorial):
mu:=ct(SUMMAND,ORDER,k,n,N):
 
if mu=0 then
 RETURN(0):
fi:
 
ope:=mu[1]:
G1:=mu[2]*SUMMAND:
gu:=0:
 
for i from 0 to degree(ope,N) do
gu:=gu+coeff(ope,N,i)*simplify1(SUMMAND,n,i):
gu:=normal(gu):
od:
 
lu:=subs(k=k+1,mu[2])*simplify1(SUMMAND,k,1)-mu[2]:
lu:=normal(lu):
normal(gu/lu);
end:
 
 
findrec:=proc(f,DEGREE,ORDER,n,N)
local ope,var,eq,a,i,j,n0,kv,var1,eq1,mu:
if (1+DEGREE)*(1+ORDER)+2+ORDER>nops(f) then
ERROR(`Insufficient data for a recurrence of order`,ORDER, `degree`,DEGREE):
fi:
ope:=0:
var:={}:
 
for i from 0 to ORDER do
 for j from 0 to DEGREE do
  ope:=ope+a[i,j]*n^j*N^i:
  var:=var union {a[i,j]}:
 od:
od:
    
eq:={}:
 
for n0 from 1 to (1+DEGREE)*(1+ORDER)+2 do
  eq1:=0:
 
  for i from 0 to ORDER do
     eq1:=eq1+subs(n=n0,coeff(ope,N,i))*op(n0+i,f):
  od:
 
   eq:= eq union {eq1}:
od:
 
var1:=solve1(eq,var):
 
kv:={}:
 
for i from 1 to nops(var1) do
 mu:=op(i,var1):
 
if op(1,mu)=op(2,mu) then
   kv:= kv union {op(1,mu)}:
 fi:
 
od:
ope:=subs(var1,ope):
 
 
if nops(kv)>1 then
  print(` either DEGREE or ORDER are too high`):
   print(`The output is not the minimal possible operator`):
fi:
 
for i from 1 to nops(kv) do
  ope:=subs(op(i,kv)=1,ope):
od:
 
ope:
 
end:
        
celine:=proc(f,ii,jj)
local m,j,l,i,tt,yy,zz,rr,xx,ss,zy,
 b, iii, k, n, r, zzz:
 
m:='m':j:='j':l:='l':i:='i':
iii:=ii*(jj+1)+jj:
xx:=seq(b[i],i=0..iii):i:='i':
ss:=numer(simplify(expand(sum(sum(b[i*(jj+1)+j]*f(n-j,k-i)/f(n,k), 
            i=0..ii),j=0..jj)))):
tt:=coeffs(collect(ss,k),k):
yy:=solve1({tt},{xx}):
zz:=simplify(sum(sum(b[l*(jj+1)+m]*F(n-m,k-l),l=0..ii),m=0..jj)): 
i:='i':j:='j':print(` ` ):print(` `):print(`The full  recurrence is`):
print(` `):
zz:=numer(simplify(subs(yy,zz))):
for i from 0 to ii do for j from 0 to jj do                
zz:=collect(zz,F(n-j,k-i)) od od:
print(zz,`==0`);zzz:=zz:
l:='l':j:='j':m:='m':
r:=subs(yy,simplify(expand(sum(sum(sum(b[l*(jj+1)+m],l=j+1..ii)*simplify(expand(f(n-m,k-j)/f(n,k))),j=0..ii),m=0..jj)))): 
 
rr:=simplify(factor(simplify(r))):print(` ` ):print(` `):
print(`The telescoped form is`);print(` `);
zy:=factor(subs(yy,sum(simplify(sum(b[l*(jj+1)+m],l=0..ii))*F(n-m,k),m=0..jj))):
print(zy,`==G(n,k)-G(n,k-1)`); 
 
print(` `);
print(`where G(n,k)=R(n,k)*F(n,k) and the rational function  R(n,k) is `):
print(` `);
rr;
end:
 
findgQR:=proc(Q,R,k,L)
local j,g:
 
for j from 1 to L do
 g:=gcd(Q,subs(k=k+j,R)):
 if degree(g,k)>0 then
  RETURN(g,j):
 fi:
od:
1,0:
end:
 
  
 
 
tovu:=proc(SU,k,n)
local gu:
 
gu:=simplify1(simplify1(SU,n,1),k,1):
 
if gu=1 then
RETURN(0):
else
RETURN(1):
fi:
end:
 
#TerryTao(P,Q,x,z,k,L,K): inputs hyper-exponential functions P and Q of the variable x, 
#a continuous variable z (for the certificate),
#a discrete variable
#k, and an affine-linear expression L in k of the form a0+b0*k for some non-negative integers a0 and b0
#and a symbol,K, for the shift operator in k.
#Outputs a recurrence, and a certificate (using the Almkvist-Zeilberger algorithm) for the
#sequence , in k, of the functions (of x)
#D_x^L(Q(x)*P(x)^k). For example, for the problem in Terry Tao's blog, May 30, 2015
#A differentiation identity, type
TerryTao(1+x^2,(1+x^2)^(-1/2),x,z,k,2*k,K);

TerryTao:=proc(P,Q,x,z,k,L,K) :

AZd(L!*subs(x=z,P)^k*subs(x=z,Q)/(z-x)^(L+1),z,k,K):


end:




####start Dec. 11, 2015 paper
AZdI:= proc(A,y,n,N,a,b)
local ORDER,gu,KAMA,ope,cert,yemin1:
KAMA:=12:
 
for ORDER from 0 to KAMA do
gu:=duisd(A,ORDER,y,n,N):
if gu<>0 then

ope:=gu[1]:
cert:=gu[2]:
yemin1:=normal(subs(y=b, normal(cert*A))-subs(y=a, normal(cert*A))):
RETURN([[ope,yemin1],cert]):

fi:
od:
0:
end:



AZcI:=proc(A,y,x,D,a,b) local ORDER,gu,KAMA,ope,cert,yemin1:
KAMA:=8:
 
for ORDER from 1 to KAMA do
gu:=duisc(A,ORDER,y,x,D):
if gu<>0 and gu[1]<>0 then
ope:=gu[1]:
cert:=gu[2]:
yemin1:=normal(subs(y=b, normal(cert*A))-subs(y=a, normal(cert*A))):
RETURN([[ope,yemin1],cert]):

fi:
od:
0:
end:




#OperZeil(F,k,n,N): Same as zeil(F,k,n,N)[1] but an operator, ope, in decreasing powers of N, such that it is equivalent to 1-ope
OperZeil:=proc(F,k,n,N) local ope,d,i,k1,n1:
option remember:
ope:=zeil(F,k,n,N,20)[1]:
d:=degree(ope,N):
ope:=expand(subs(n=n-d,ope)/N^d):
ope:=1-ope/coeff(ope,N,0):
[add(factor(coeff(ope,N,-i))/N^i,i=1..d),[seq(add(eval(subs({n=n1,k=k1},F)),k1=0..n1),n1=0..d-1)]]:
end:


#OperAZd(F,x,n,N): Same as AZe(F,x,n,N)[1] but an operator, ope, in decreasing powers of N, such that it is equivalent to 1-ope
OperAZd:=proc(F,x,n,N) local ope,d,i:
option remember:
ope:=AZd(F,x,n,N,20)[1]:
d:=degree(ope,N):
ope:=expand(subs(n=n-d,ope)/N^d):
ope:=1-ope/coeff(ope,N,0):
add(factor(coeff(ope,N,-i))/N^i,i=1..d):
end:


#SeqFromRec(ope,n,N,Ini,n0,K): inputs an operator given in temrs of the negative shift operator
#1/N, and a list Ini, givining the values at n=n0, ...,n=n0+L-1, finds the list from n=1 to
#to n=K. Try:
#SeqFromRec(1/N+1/N^2,n,N,[0,1],0,20);
SeqFromRec:=proc(ope,n,N,Ini,n0,K) local d,i,T,n1,lu:

lu:=denom(ope):

d:=-ldegree(ope,N):


if member(0,{seq(subs(n=n1,lu),n1=n0+nops(Ini)..K)}) then
 RETURN(FAIL):
fi:



for n1 from 0 to nops(Ini)-1 do
 T[n0+n1]:=Ini[n1+1]:
od:

for n1 from n0+nops(Ini) to K do
 T[n1]:=add( subs(n=n1,coeff(ope,N,-i))*T[n1-i],i=1..d):
od:

[seq(T[n1],n1=1..K)]:

end:
 



#RAT(F,k,n,K): inputs a closed-form expression and outputs  a(K)/b(K) where
#b(n):=Sum(F,k=0..n) and a(n) is satisfies the same initial conditions  with a(i)=0  for negative i and 0 and a(1)=1, Try
#RAT(binomial(n,k)^3,k,n,1000);
RAT:=proc(F,k,n,K) local ope,N,d,Ini1,Ini2,lu1,lu2,k1,n1:
 ope:=OperZeil(F,k,n,N)[1]:
 d:=-ldegree(ope,N):
if d<2 then
 RETURN(FAIL):
fi:

Ini2:=[seq(add(eval(subs({n=n1,k=k1},F)),k1=0..n1),n1=0..d-1)]:
lu2:=SeqFromRec(ope,n,N,Ini2,0,K):

if lu2=FAIL then
 RETURN(FAIL):
fi:

 Ini1:=[0$(d-2),0,1]:

lu1:=SeqFromRec(ope,n,N,Ini1,2-d,K):
lu1[K]/lu2[K]:
end:


#RATaz(P,x,K): inputs a Laurent polynoimal P in the cont. variable x and the discrete variable n
#and outputs  a(K)/b(K) where
#b(n):=CT(P^n,x) and a(n) is satisfies the same initial conditions  with a(i)=0  for negative i and 0 and a(1)=1, Try
#RATaz(1/x+1+x,x,1000);
RATaz:=proc(P,x,K) local n,ope,N,d,Ini1,Ini2,lu1,lu2,n1:
 ope:=OperAZd(P^n/x,x,n,N):

 d:=-ldegree(ope,N):
if d<2 then
 RETURN(FAIL):
fi:

Ini2:=[seq(coeff(P^n1,x,0),n1=0..d-1)]:
lu2:=SeqFromRec(ope,n,N,Ini2,0,K):


 Ini1:=[0$(d-2),0,1]:

lu1:=SeqFromRec(ope,n,N,Ini1,2-d,K):
lu1[K]/lu2[K]:
end:




#DEL(F,k,n,K): inputs a closed-form expression and outputs  the sequence a(n)/b(n) where
#b(n):=Sum(F,k=0..n) and a(n) is satisfies the same initial conditions  with a(i)=0  for negative i and 0 and a(1)=1, 
#It also returns the deltas. Try:
#DEL(binomial(n,k)*binomial(n+k,k)*binomial(n+2*k,k),k,n,1000);
#DEL(binomial(n,k)*binomial(n+k,k)*binomial(n+2*k,k)*a^k,k,n,1000), for a=2,3,4,

DEL:=proc(F,k,n,K) local ope,N,d,Ini1,Ini2,lu1,lu2,k1,n1,i,C,S,mu,mu1:
 ope:=OperZeil(F,k,n,N)[1]:
 d:=-ldegree(ope,N):

if K<100 then
  print(K, `must be at least 100`):
  RETURN(FAIL):
fi:

if d<2 then
 RETURN(FAIL):
fi:

Ini2:=[seq(add(eval(subs({n=n1,k=k1},F)),k1=0..n1),n1=0..d-1)]:
lu2:=SeqFromRec(ope,n,N,Ini2,0,2*K):

if lu2=FAIL then
 RETURN(FAIL):
fi:


 Ini1:=[0$(d-2),0,1]:

lu1:=SeqFromRec(ope,n,N,Ini1,2-d,2*K):


if lu1=FAIL then
 RETURN(FAIL):
fi:

S:=[seq(lu1[i]/lu2[i],i=1..2*K)]:

C:=S[2*K]:


mu:=evalf(C):

    mu1:=MyIDsLog(mu,30,10000):
      if mu1=FAIL then
         mu:=identify(mu):
       else
       mu:=mu1:
      fi:
mu,min(seq(evalf(-log(abs(C-S[i]))/log(denom(S[i])))-1,i=K..K+10)):

end:



#DELaz(P,x,K): inputs a Laurent polynoimal P in the cont. variable x and the discrete variable n
#and outputs  a(K)/b(K) where
#b(n):=CT(P^n,x) and a(n) is satisfies the same initial conditions  with a(i)=0  for negative i and 0 and a(1)=1, Try
#DELaz(1/x+1+x,x,1000);
DELaz:=proc(P,x,K) local n,N,ope,d,Ini1,Ini2,lu1,lu2,i,n1,S,C,mu,mu1:
 ope:=OperAZd(P^n/x,x,n,N):
 d:=-ldegree(ope,N):
if d<2 then
 RETURN(FAIL):
fi:

Ini2:=[seq(coeff(P^n1,x,0),n1=0..d-1)]:
lu2:=SeqFromRec(ope,n,N,Ini2,0,2*K):

if lu2=FAIL then
 RETURN(FAIL):
fi:

 Ini1:=[0$(d-2),0,1]:

lu1:=SeqFromRec(ope,n,N,Ini1,2-d,2*K):


if lu1=FAIL then
 RETURN(FAIL):
fi:

S:=[seq(lu1[i]/lu2[i],i=1..2*K)]:

C:=S[2*K]:

mu:=evalf(C):

    mu1:=MyIDsLog(mu,30,10000):
      if mu1=FAIL then
         mu:=identify(mu):
       else
       mu:=mu1:
      fi:
mu,min(seq(evalf(-log(abs(C-S[i]))/log(denom(S[i])))-1,i=K..K+10)):

end:





#MyIDs(C,F,F0,N): Given a constant C in decimals and another constant
#let's call it F, whose floating-point if F0
# and a positive integer N
#tries to express C as (a*F+b)/(c*F+d) for a,b,c,d from -N to N using PSLQ
#MyIDs(evalf((log(2)-2)/(2*log(2)+3)),log(2),evalf(log(2)),100);
MyIDs:=proc(C,F,F0,N) local gu,mu,i:

gu:=IntegerRelations[PSLQ]([1,evalf(C),evalf(F0),evalf(C*F0)]):


if max(seq(abs(gu[i]),i=1..4)) >N then
 RETURN(FAIL):
fi:

if gu[2]+gu[4]*F0=0 then
 RETURN(FAIL):
fi:


mu:=-(gu[1]+gu[3]*F)/(gu[2]+gu[4]*F):


if abs(evalf(subs(F=F0,mu)-C))>1/10^(Digits-7) then
  RETURN(FAIL):
fi:

mu:

end:


#MyIDsLog(C,K,N): guesses a fractional-linear  expression for the constant C with coefficients <=N in aba. value (given in floating point) in terms of log(j/i) where 2<=i<j<=K. Try:
#MyIDsLog( evalf( (log(5)-log(2))/(log(5)+log(2))) ,5,1000);
MyIDsLog:=proc(C,K,N) local F,i,j,gu:
for i from 2 to K do
 for j from 1 to i-1 do


 gu:=MyIDs(C,F,evalf(log(i/j)),N):


 if gu<>FAIL then
   RETURN(simplify(subs(F=log(i/j),gu))):
fi:

od:

od:

FAIL:

end:


#FindAZ1(x,A,K): Inputs a positive integer d, a variable x, a positive integer A and a large positive integer K
#outputs the list of pairs [P,delta] such that P is a Laurent polynomial of low-degree -1, degree 1, and the coefficients
#are positive integers between 1 and A, and the implied delta (using s sequence of length 2*K), in the Apery limit of the recurrence
#satisfied by the constant term of P^n  yields a positive delta, followed by a conjectured value (produced by
#Maple, identified or own, if known. Try:
#FindAZ1(x,3,300);
FindAZ1:=proc(x,A,K) local L,a1,a2,a3,gu,P:
L:=[]:

for a1 from 1 to A do
 for a2 from 1 to A do
  for a3 from a1 to A do

   if igcd(a1,a2,a3)=1 then

   P:=a1/x+a2+a3*x:
 
    gu:=DELaz(P,x,K):

 if gu<>FAIL then
    if gu[2]>0 then

      L:=[op(L),[factor(P),gu]]:

   fi:
fi:
fi:

  od:
 od:
od:

L:

end:


#FindAZ2(x,A,K): Inputs a positive integer d, a variable x, a positive integer A and a large positive integer K
#outputs the list of pairs [P,delta] such that P is a Laurent polynomial of low-degree -2, degree 2, and the coefficients
#are positive integers between 1 and A, and the implied delta (using s sequence of length 2*K), in the Apery limit of the recurrence
#satisfied by the constant term of P^n  yields a positive delta, followed by a conjectured value (produced by
#Maple, identified or own, if known. Try:
#FindAZ2(x,3,300);
FindAZ2:=proc(x,A,K) local L,a1,a2,a3,a4,a5,gu,P,mu:
L:=[]:

for a1 from 1 to A do
 for a2 from 1 to A do
  for a3 from 1 to A do
   for a4 from 1 to A do
    for a5 from 1 to A do
   P:=a1/x^2+a2/x+a3+a4*x+a5*x^2:
 
    gu:=DELaz(P,x,K):

 if gu<>FAIL then
    if gu[2]>0 then

    mu:=MyIDsLog(gu[1],20,1000):
      if mu=FAIL then
         mu:=identify(gu[1]):
      fi:

      L:=[op(L),[factor(P),mu,gu[2]]]:

   fi:
fi:
  od:
 od:
od:
od:
od:

L:

end:




    

#FindZ1(A,n,k,K): Inputs a positive  integer A and a large positive integer K
#outputs the list of pairs [F,DEL(F,k,n,K)] such that F is an expression of the form binomial(n,k)*binomial(a1*n+b1*k,a2*n+b2*k)*x^k
#where a1 is from 1 to A and b1 from 0 to A and a2<=a1 and b2<=b1 and x is between 1 and A
#such that DEL(F,k,n,K)[2] is positive. Try:
#FindZ1(1,n,k,300);
FindZ1:=proc(A,n,k,K) local a1,a2,b1,b2,x,F,L,gu:
L:=[]:

for a1 from 1 to A do
 for a2 from 0 to A do
  for b1 from 1 to a1 do
   for b2 from 0 to a2 do

    if  b1<a1 or b2<a2 then

     for x from 1 to A do
      F:=binomial(n,k)*binomial(a1*n+a2*k,b1*n+b2*k)*x^k:
  
        gu:=DEL(F,k,n,K):

       if gu<>FAIL then
          if gu[2]>0 then
          
           L:= [op(L),[F,gu]]:

          fi:

    fi:
   od:
 fi: 
od:
od:
od:
od:
L:
end:





    

#FindZ2(A,n,k,K): Inputs a positive  integer A and a large positive integer K
#outputs the list of pairs [F,DEL(F,k,n,K)] such that F is an expression of the form binomial(n,k)*binomial(a1*n+b1*k,k)*binomial(a2*n+b2*k,k)*x^k
#where a1 is from 1 to A and b1 from 0 to A and a2<=a1 and b2<=b1 and x is between 1 and A
#such that DEL(F,k,n,K)[2] is positive. Try:
#FindZ2(1,n,k,300);
FindZ2:=proc(A,n,k,K) local a1,a2,b1,b2,x,F,L,gu,Done1:
L:=[]:

Done1:={}:
for a1 from 1 to A do
 for a2 from 1 to A do
  for b1 from 0 to A do
   for b2 from 0 to A do
     for x from 1 to A do
      F:=binomial(n,k)*binomial(a1*n+b1*k,k)*binomial(a2*n+b2*k,k)*x^k:
    
   if not member(F,Done1) then
       Done1:=Done1 union {F}:

    
        gu:=DEL(F,k,n,K):

       if gu<>FAIL then
          if gu[2]>0 then
          
           L:= [op(L),[F,gu]]:

          fi:
      fi:
fi:
   od:
od:
od:
od:
od:
L:
end:


#DELazG(F,x,n,K): inputs a hyperexponential function F (cont. in x, discrete in n) in the cont. variable x and the discrete variable n
#and outputs  a(K)/b(K) where a(n),b(n) are solutions of the recurrence satisfied by the controur integral of F around the origin
#with initial conditions [1,0,...0] and [0,1,...] respectively. It also gives you the estimated delta of the sequence. Try:
#DELazG((1/x+1+x,x)^n/x,x,n,1000);
DELazG:=proc(F,x,n,K) local N,ope,d,Ini1,Ini2,lu1,lu2,i,S,C,mu,mu1:
 ope:=OperAZd(F,x,n,N):
 d:=-ldegree(ope,N):
if d<2 then
 RETURN(FAIL):
fi:

Ini2:=[1,0$(d-1)]:
lu2:=SeqFromRec(ope,n,N,Ini2,0,2*K):


if lu2=FAIL then
 RETURN(FAIL):
fi:

 Ini1:=[0,1,0$(d-2)]:

lu1:=SeqFromRec(ope,n,N,Ini1,2-d,2*K):

if lu1=FAIL then
 RETURN(FAIL):
fi:

S:=[seq(lu1[i]/lu2[i],i=d+1..2*K)]:

C:=S[-1]:


if C=0 then
 RETURN(FAIL):
fi:

mu:=evalf(C):

    mu1:=MyIDsLog(mu,30,10000):
      if mu1=FAIL then
         mu:=identify(mu):
       else
       mu:=mu1:
      fi:
mu,min(seq(evalf(-log(abs(C-S[i]))/log(denom(S[i])))-1,i=K..K+10)):

end:





#DELaz(P,x,K): inputs a Laurent polynoimal P in the cont. variable x and the discrete variable n
#and outputs  a(K)/b(K) where
#b(n):=CT(P^n,x) and a(n) is satisfies the same initial conditions  with a(i)=0  for negative i and 0 and a(1)=1, Try
#DELaz(1/x+1+x,x,1000);
DELaz:=proc(P,x,K) local n,N,ope,d,Ini1,Ini2,lu1,lu2,i,n1,S,C,mu,mu1:
 ope:=OperAZd(P^n/x,x,n,N):
 d:=-ldegree(ope,N):

if d<2 then
 RETURN(FAIL):
fi:




Ini2:=[seq(normal(P^n1),n1=0..d-1)]:

Ini2:=[seq(coeff(series(Ini2[i],x=0),x,0), i=1..nops(Ini2))]:


lu2:=SeqFromRec(ope,n,N,Ini2,0,2*K):

if lu2=FAIL then
 RETURN(FAIL):
fi:

 Ini1:=[0$(d-2),0,1]:

lu1:=SeqFromRec(ope,n,N,Ini1,2-d,2*K):


if lu1=FAIL then
 RETURN(FAIL):
fi:

S:=[seq(lu1[i]/lu2[i],i=1..2*K)]:

C:=S[2*K]:

mu:=evalf(C):

    mu1:=MyIDsLog(mu,30,10000):
      if mu1=FAIL then
         mu:=identify(mu):
       else
       mu:=mu1:
      fi:
mu,min(seq(evalf(-log(abs(C-S[i]))/log(denom(S[i])))-1,i=K..K+10)):

end:





#MyIDs(C,F,F0,N): Given a constant C in decimals and another constant
#let's call it F, whose floating-point if F0
# and a positive integer N
#tries to express C as (a*F+b)/(c*F+d) for a,b,c,d from -N to N using PSLQ
#MyIDs(evalf((log(2)-2)/(2*log(2)+3)),log(2),evalf(log(2)),100);
MyIDs:=proc(C,F,F0,N) local gu,mu,i:

gu:=IntegerRelations[PSLQ]([1,evalf(C),evalf(F0),evalf(C*F0)]):


if max(seq(abs(gu[i]),i=1..4)) >N then
 RETURN(FAIL):
fi:

mu:=-(gu[1]+gu[3]*F)/(gu[2]+gu[4]*F):

if abs(evalf(subs(F=F0,mu)-C))>1/10^(Digits-7) then
  RETURN(FAIL):
fi:

mu:

end:


#MyIDsLog(C,K,N): guesses a fractional-linear  expression for the constant C with coefficients <=N in aba. value (given in floating point) in terms of log(j/i) where 2<=i<j<=K. Try:
#MyIDsLog( evalf( (log(5)-log(2))/(log(5)+log(2))) ,5,1000);
MyIDsLog:=proc(C,K,N) local F,i,j,gu:
for i from 2 to K do
 for j from 1 to i-1 do


 gu:=MyIDs(C,F,evalf(log(i/j)),N):


 if gu<>FAIL then
   RETURN(simplify(subs(F=log(i/j),gu))):
fi:

od:

od:

FAIL:

end:


#FindAZ1(x,A,K): Inputs a positive integer d, a variable x, a positive integer A and a large positive integer K
#outputs the list of pairs [P,delta] such that P is a Laurent polynomial of low-degree -1, degree 1, and the coefficients
#are positive integers between 1 and A, and the implied delta (using s sequence of length 2*K), in the Apery limit of the recurrence
#satisfied by the constant term of P^n  yields a positive delta, followed by a conjectured value (produced by
#Maple, identified or own, if known. Try:
#FindAZ1(x,3,300);
FindAZ1:=proc(x,A,K) local L,a1,a2,a3,gu,P:
L:=[]:

for a1 from 1 to A do
 for a2 from 1 to A do
  for a3 from a1 to A do

   if igcd(a1,a2,a3)=1 then

   P:=a1/x+a2+a3*x:
 
    gu:=DELaz(P,x,K):

 if gu<>FAIL then
    if gu[2]>0 then

      L:=[op(L),[factor(P),gu]]:

   fi:
fi:
fi:

  od:
 od:
od:

L:

end:


#FindAZ2(x,A,K): Inputs a positive integer d, a variable x, a positive integer A and a large positive integer K
#outputs the list of pairs [P,delta] such that P is a Laurent polynomial of low-degree -2, degree 2, and the coefficients
#are positive integers between 1 and A, and the implied delta (using s sequence of length 2*K), in the Apery limit of the recurrence
#satisfied by the constant term of P^n  yields a positive delta, followed by a conjectured value (produced by
#Maple, identified or own, if known. Try:
#FindAZ2(x,3,300);
FindAZ2:=proc(x,A,K) local L,a1,a2,a3,a4,a5,gu,P,mu:
L:=[]:

for a1 from 1 to A do
 for a2 from 1 to A do
  for a3 from 1 to A do
   for a4 from 1 to A do
    for a5 from 1 to A do
   P:=a1/x^2+a2/x+a3+a4*x+a5*x^2:
 
    gu:=DELaz(P,x,K):

 if gu<>FAIL then
    if gu[2]>0 then

    mu:=MyIDsLog(gu[1],20,1000):
      if mu=FAIL then
         mu:=identify(gu[1]):
      fi:

      L:=[op(L),[factor(P),mu,gu[2]]]:

   fi:
fi:
  od:
 od:
od:
od:
od:

L:

end:




    


#DELazG(F,x,n,K): inputs a hyperexponential function F (cont. in x, discrete in n) in the cont. variable x and the discrete variable n
#and outputs  a(K)/b(K) where a(n),b(n) are solutions of the recurrence satisfied by the controur integral of F around the origin
#with initial conditions [1,0,...0] and [0,1,...] respectively. It also gives you the estimated delta of the sequence. Try:
#DELazG((1/x+1+x,x)^n/x,x,n,1000);
DELazG:=proc(F,x,n,K) local N,ope,d,Ini1,Ini2,lu1,lu2,i,S,C,mu,mu1:
 ope:=OperAZd(F,x,n,N):
 d:=-ldegree(ope,N):
if d<2 then
 RETURN(FAIL):
fi:

Ini2:=[1,0$(d-1)]:
lu2:=SeqFromRec(ope,n,N,Ini2,0,2*K):


if lu2=FAIL then
 RETURN(FAIL):
fi:

 Ini1:=[0,1,0$(d-2)]:

lu1:=SeqFromRec(ope,n,N,Ini1,2-d,2*K):

if lu1=FAIL then
 RETURN(FAIL):
fi:

S:=[seq(lu1[i]/lu2[i],i=d+1..2*K)]:

C:=S[-1]:


if C=0 then
 RETURN(FAIL):
fi:

mu:=evalf(C):

    mu1:=MyIDsLog(mu,30,10000):
      if mu1=FAIL then
         mu:=identify(mu):
       else
       mu:=mu1:
      fi:
mu,min(seq(evalf(-log(abs(C-S[i]))/log(denom(S[i])))-1,i=K..K+10)):

end:

#OpeMAZ1(a,b,c,n,N): The linear recurrence operator in n and the shift operator N satisfied by the constant term of In z, w) of
#((1+z)*(1+w)*(a*w+b*z+c*w*z)/w/z)^n. In other words the output of
#MAZ(1,1/z/w,(1+z)*(1+w)*(a*w+b*z+c*w*z)/w/z,[w,z],n,N,[a,b,c])[1]`): in the Maple package MAZ.txt. Try:
#OpeMAZ1(1,1,1,n,N);
OpeMAZ1:=proc(a,b,c,n,N) local ope,i:
ope:=
-(n+1)^2*(-c+b)*(a-c)*(a+b-c)^3*(27*a^2*b^2*n^2-288*a*b*c^2*n^2+256*a*c^3*n^2+256*b*c^3*n^2-256*c^4*n^2+171*a^2*b^2*n-24*a^2*b*c*n-24*a*b^2*c*n-1620*a*b*c^2*n+1472*
a*c^3*n+1472*b*c^3*n-1472*c^4*n+252*a^2*b^2-66*a^2*b*c-66*a*b^2*c-2202*a*b*c^2+2052*a*c^3+2052*b*c^3-2052*c^4)/(n+4)^2/(27*a^2*b^2*n^2-288*a*b*c^2*n^2+256*a*c^3*n^2
+256*b*c^3*n^2-256*c^4*n^2+117*a^2*b^2*n-24*a^2*b*c*n-24*a*b^2*c*n-1044*a*b*c^2*n+960*a*c^3*n+960*b*c^3*n-960*c^4*n+108*a^2*b^2-42*a^2*b*c-42*a*b^2*c-870*a*b*c^2+
836*a*c^3+836*b*c^3-836*c^4)/c+(81*a^5*b^3*n^4-108*a^5*b^2*c*n^4-567*a^4*b^4*n^4+486*a^4*b^3*c*n^4-756*a^4*b^2*c^2*n^4+1920*a^4*b*c^3*n^4-1024*a^4*c^4*n^4+81*a^3*b^
5*n^4+486*a^3*b^4*c*n^4+5373*a^3*b^3*c^2*n^4-9684*a^3*b^2*c^3*n^4+1664*a^3*b*c^4*n^4+2048*a^3*c^5*n^4-108*a^2*b^5*c*n^4-756*a^2*b^4*c^2*n^4-9684*a^2*b^3*c^3*n^4+
21684*a^2*b^2*c^4*n^4-11136*a^2*b*c^5*n^4+1920*a*b^4*c^3*n^4+1664*a*b^3*c^4*n^4-11136*a*b^2*c^5*n^4+9600*a*b*c^6*n^4-2048*a*c^7*n^4-1024*b^4*c^4*n^4+2048*b^3*c^5*n^
4-2048*b*c^7*n^4+1024*c^8*n^4+756*a^5*b^3*n^3-1080*a^5*b^2*c*n^3+96*a^5*b*c^2*n^3-5292*a^4*b^4*n^3+4968*a^4*b^3*c*n^3-6780*a^4*b^2*c^2*n^3+16560*a^4*b*c^3*n^3-8960*
a^4*c^4*n^3+756*a^3*b^5*n^3+4968*a^3*b^4*c*n^3+45000*a^3*b^3*c^2*n^3-83520*a^3*b^2*c^3*n^3+14608*a^3*b*c^4*n^3+17920*a^3*c^5*n^3-1080*a^2*b^5*c*n^3-6780*a^2*b^4*c^2
*n^3-83520*a^2*b^3*c^3*n^3+188580*a^2*b^2*c^4*n^3-97200*a^2*b*c^5*n^3+96*a*b^5*c^2*n^3+16560*a*b^4*c^3*n^3+14608*a*b^3*c^4*n^3-97200*a*b^2*c^5*n^3+83856*a*b*c^6*n^3
-17920*a*c^7*n^3-8960*b^4*c^4*n^3+17920*b^3*c^5*n^3-17920*b*c^7*n^3+8960*c^8*n^3+2466*a^5*b^3*n^2-3723*a^5*b^2*c*n^2+576*a^5*b*c^2*n^2-17343*a^4*b^4*n^2+17325*a^4*b
^3*c*n^2-21513*a^4*b^2*c^2*n^2+51120*a^4*b*c^3*n^2-28112*a^4*c^4*n^2+2466*a^3*b^5*n^2+17325*a^3*b^4*c*n^2+136542*a^3*b^3*c^2*n^2-260073*a^3*b^2*c^3*n^2+46720*a^3*b*
c^4*n^2+56224*a^3*c^5*n^2-3723*a^2*b^5*c*n^2-21513*a^2*b^4*c^2*n^2-260073*a^2*b^3*c^3*n^2+590637*a^2*b^2*c^4*n^2-305328*a^2*b*c^5*n^2+576*a*b^5*c^2*n^2+51120*a*b^4*
c^3*n^2+46720*a*b^3*c^4*n^2-305328*a*b^2*c^5*n^2+263136*a*b*c^6*n^2-56224*a*c^7*n^2-28112*b^4*c^4*n^2+56224*b^3*c^5*n^2-56224*b*c^7*n^2+28112*c^8*n^2+3357*a^5*b^3*n
-5301*a^5*b^2*c*n+1092*a^5*b*c^2*n-23958*a^4*b^4*n+25065*a^4*b^3*c*n-28719*a^4*b^2*c^2*n+66984*a^4*b*c^3*n-37528*a^4*c^4*n+3357*a^3*b^5*n+25065*a^3*b^4*c*n+178401*a
^3*b^3*c^2*n-347967*a^3*b^2*c^3*n+65096*a^3*b*c^4*n+75056*a^3*c^5*n-5301*a^2*b^5*c*n-28719*a^2*b^4*c^2*n-347967*a^2*b^3*c^3*n+792555*a^2*b^2*c^4*n-410568*a^2*b*c^5*
n+1092*a*b^5*c^2*n+66984*a*b^4*c^3*n+65096*a*b^3*c^4*n-410568*a*b^2*c^5*n+352452*a*b*c^6*n-75056*a*c^7*n-37528*b^4*c^4*n+75056*b^3*c^5*n-75056*b*c^7*n+37528*c^8*n+
1620*a^5*b^3-2658*a^5*b^2*c+666*a^5*b*c^2-11880*a^4*b^4+12906*a^4*b^3*c-13662*a^4*b^2*c^2+31476*a^4*b*c^3-18036*a^4*c^4+1620*a^3*b^5+12906*a^3*b^4*c+84924*a^3*b^3*c
^2-169506*a^3*b^2*c^3+33552*a^3*b*c^4+36072*a^3*c^5-2658*a^2*b^5*c-13662*a^2*b^4*c^2-169506*a^2*b^3*c^3+386094*a^2*b^2*c^4-200268*a^2*b*c^5+666*a*b^5*c^2+31476*a*b^
4*c^3+33552*a*b^3*c^4-200268*a*b^2*c^5+170646*a*b*c^6-36072*a*c^7-18036*b^4*c^4+36072*b^3*c^5-36072*b*c^7+18036*c^8)/(n+4)^2/(27*a^2*b^2*n^2-288*a*b*c^2*n^2+256*a*c
^3*n^2+256*b*c^3*n^2-256*c^4*n^2+117*a^2*b^2*n-24*a^2*b*c*n-24*a*b^2*c*n-1044*a*b*c^2*n+960*a*c^3*n+960*b*c^3*n-960*c^4*n+108*a^2*b^2-42*a^2*b*c-42*a*b^2*c-870*a*b*
c^2+836*a*c^3+836*b*c^3-836*c^4)/c*N-(81*a^4*b^3*n^4-162*a^4*b^2*c*n^4+81*a^3*b^4*n^4+567*a^3*b^3*c*n^4-972*a^3*b^2*c^2*n^4+2496*a^3*b*c^3*n^4-1536*a^3*c^4*n^4-162*
a^2*b^4*c*n^4-972*a^2*b^3*c^2*n^4-4674*a^2*b^2*c^3*n^4+4224*a^2*b*c^4*n^4+512*a^2*c^5*n^4+2496*a*b^3*c^3*n^4+4224*a*b^2*c^4*n^4-5696*a*b*c^5*n^4-512*a*c^6*n^4-1536*
b^3*c^4*n^4+512*b^2*c^5*n^4-512*b*c^6*n^4+1536*c^7*n^4+837*a^4*b^3*n^3-1746*a^4*b^2*c*n^3+144*a^4*b*c^2*n^3+837*a^3*b^4*n^3+5715*a^3*b^3*c*n^3-9792*a^3*b^2*c^2*n^3+
24216*a^3*b*c^3*n^3-14976*a^3*c^4*n^3-1746*a^2*b^4*c*n^3-9792*a^2*b^3*c^2*n^3-44718*a^2*b^2*c^3*n^3+41184*a^2*b*c^4*n^3+4992*a^2*c^5*n^3+144*a*b^4*c^2*n^3+24216*a*b
^3*c^3*n^3+41184*a*b^2*c^4*n^3-55752*a*b*c^5*n^3-4992*a*c^6*n^3-14976*b^3*c^4*n^3+4992*b^2*c^5*n^3-4992*b*c^6*n^3+14976*c^7*n^3+3096*a^4*b^3*n^2-6717*a^4*b^2*c*n^2+
1044*a^4*b*c^2*n^2+3096*a^3*b^4*n^2+21054*a^3*b^3*c*n^2-36174*a^3*b^2*c^2*n^2+85356*a^3*b*c^3*n^2-53080*a^3*c^4*n^2-6717*a^2*b^4*c*n^2-36174*a^2*b^3*c^2*n^2-155325*
a^2*b^2*c^3*n^2+146916*a^2*b*c^4*n^2+17352*a^2*c^5*n^2+1044*a*b^4*c^2*n^2+85356*a*b^3*c^3*n^2+146916*a*b^2*c^4*n^2-199836*a*b*c^5*n^2-17352*a*c^6*n^2-53080*b^3*c^4*
n^2+17352*b^2*c^5*n^2-17352*b*c^6*n^2+53080*c^7*n^2+4842*a^4*b^3*n-10893*a^4*b^2*c*n+2436*a^4*b*c^2*n+4842*a^3*b^4*n+33312*a^3*b^3*c*n-57762*a^3*b^2*c^2*n+128760*a^
3*b*c^3*n-80632*a^3*c^4*n-10893*a^2*b^4*c*n-57762*a^2*b^3*c^2*n-230817*a^2*b^2*c^3*n+226476*a^2*b*c^4*n+25256*a^2*c^5*n+2436*a*b^4*c^2*n+128760*a*b^3*c^3*n+226476*a
*b^2*c^4*n-309776*a*b*c^5*n-25256*a*c^6*n-80632*b^3*c^4*n+25256*b^2*c^5*n-25256*b*c^6*n+80632*c^7*n+2664*a^4*b^3-6192*a^4*b^2*c+1806*a^4*b*c^2+2664*a^3*b^4+18792*a^
3*b^3*c-33270*a^3*b^2*c^2+69282*a^3*b*c^3-43836*a^3*c^4-6192*a^2*b^4*c-33270*a^2*b^3*c^2-122556*a^2*b^2*c^3+126510*a^2*b*c^4+12708*a^2*c^5+1806*a*b^4*c^2+69282*a*b^
3*c^3+126510*a*b^2*c^4-174090*a*b*c^5-12708*a*c^6-43836*b^3*c^4+12708*b^2*c^5-12708*b*c^6+43836*c^7)/(n+4)^2/(27*a^2*b^2*n^2-288*a*b*c^2*n^2+256*a*c^3*n^2+256*b*c^3
*n^2-256*c^4*n^2+117*a^2*b^2*n-24*a^2*b*c*n-24*a*b^2*c*n-1044*a*b*c^2*n+960*a*c^3*n+960*b*c^3*n-960*c^4*n+108*a^2*b^2-42*a^2*b*c-42*a*b^2*c-870*a*b*c^2+836*a*c^3+
836*b*c^3-836*c^4)/c*N^2+(27*a^3*b^3*n^4-108*a^3*b^2*c*n^4-108*a^2*b^3*c*n^4-396*a^2*b^2*c^2*n^4+1408*a^2*b*c^3*n^4-1024*a^2*c^4*n^4+1408*a*b^2*c^3*n^4-1152*a*b*c^4
*n^4-1024*b^2*c^4*n^4+1024*c^6*n^4+306*a^3*b^3*n^3-1248*a^3*b^2*c*n^3+96*a^3*b*c^2*n^3-1248*a^2*b^3*c*n^3-4092*a^2*b^2*c^2*n^3+15088*a^2*b*c^3*n^3-11008*a^2*c^4*n^3
+96*a*b^3*c^2*n^3+15088*a*b^2*c^3*n^3-12528*a*b*c^4*n^3-11008*b^2*c^4*n^3+11008*c^6*n^3+1251*a^3*b^3*n^2-5205*a^3*b^2*c*n^2+816*a^3*b*c^2*n^2-5205*a^2*b^3*c*n^2-\
15285*a^2*b^2*c^2*n^2+58592*a^2*b*c^3*n^2-42832*a^2*c^4*n^2+816*a*b^3*c^2*n^2+58592*a*b^2*c^3*n^2-49600*a*b*c^4*n^2-42832*b^2*c^4*n^2+42832*c^6*n^2+2160*a^3*b^3*n-\
9159*a^3*b^2*c*n+2220*a^3*b*c^2*n-9159*a^2*b^3*c*n-24051*a^2*b^2*c^2*n+96608*a^2*b*c^3*n-70728*a^2*c^4*n+2220*a*b^3*c^2*n+96608*a*b^2*c^3*n-83924*a*b*c^4*n-70728*b^
2*c^4*n+70728*c^6*n+1296*a^3*b^3-5580*a^3*b^2*c+1878*a^3*b*c^2-5580*a^2*b^3*c-13056*a^2*b^2*c^2+56124*a^2*b*c^3-41228*a^2*c^4+1878*a*b^3*c^2+56124*a*b^2*c^3-50506*a
*b*c^4-41228*b^2*c^4+41228*c^6)/(n+4)^2/(27*a^2*b^2*n^2-288*a*b*c^2*n^2+256*a*c^3*n^2+256*b*c^3*n^2-256*c^4*n^2+117*a^2*b^2*n-24*a^2*b*c*n-24*a*b^2*c*n-1044*a*b*c^2
*n+960*a*c^3*n+960*b*c^3*n-960*c^4*n+108*a^2*b^2-42*a^2*b*c-42*a*b^2*c-870*a*b*c^2+836*a*c^3+836*b*c^3-836*c^4)/c*N^3+N^4
:

ope:=expand(subs(n=n-degree(ope,N),ope)/N^degree(ope,N)):

ope:=1-expand(ope/coeff(ope,N,0)):
add(factor(coeff(ope,N,-i))*N^(-i),i=1..-ldegree(ope,N)):

end:

with(IntegerRelations):

# GEk(k, a, x): Guess a degree k polynomial in x that
#   c = DELazG(((1 + a * x) * (1 + (a + 1) * x) / x)^n / x^(1 / k), x, n, 1000)[1]:
# seems to satisfy.
# Returns [p, p(c)].
# Try:
#   GEk(2, 2, x);
#   GEk(3, 3, x);
GEk := proc(k, a, x)
    option remember:
    local L, c, i, p, j:
    c := DELazG(((1 + a * x) * (1 + (a + 1) * x) / x)^n / x^((k - 1) / k), x, n, 1000)[1]:
    L := PSLQ([seq(c^j, j=0..k)]):
    p := add(L[i] * x^(i - 1), i=1..nops(L)):

    [p, evalf(subs(x=c, p))]:
end:

# GEkCoeff(k, j, N): Return the coefficients on x^j for the polynomials GEk(k,
# a, x) for a = 1, 2, ..., N.
GEkCoeff := proc(k, j, N)
    local a,x:
    [seq(coeff(GEk(k, a, x)[1], x, j), a=1..N)]:
end:

# guessPol: Given a list L, guess a polynomial of minimal that generates it in
# the symbol a.
guessPol := proc(L, a)
    local c, deg, p, eqns, soln, pG, k, i:
    for deg from 0 to nops(L) - 1 do
        p := add(c[k] * a^k, k=0..deg):
        eqns := {seq(L[i] - subs(a=i, p), i=1..deg + 1)}:
        soln := solve(eqns, {seq(c[i], i=0..deg)}):
        if soln <> NULL then
            pG := subs(soln, p):
            if andmap(i -> subs(a = i, pG) = L[i], [seq(1..nops(L))]) then
                return factor(pG):
            fi:
        fi:
    end:

    FAIL:
end:

# guessGEk(k, j, N): Guess a polynomial expression in a for the
# coefficients of GEk(k, a, x).
guessGEk := proc(k, a, N)
    local j:
    [seq(guessPol(GEkCoeff(k, j, N), a), j=0..k)]:
end:




#START Story procedure
#DELv(F,k,n,K): verbose form of DEL(F,k,n,K).
DELv:=proc(F,k,n,K) local ope,N,d,Ini1,Ini2,lu1,lu2,k1,n1,i,C,S,mu,mu1,b,a,delt,t0:

t0:=time():
print(``):
print(`-------------------------------------------------------------`):
print(``):
print(`The Apery limit generated by the binomial coefficient sum`, Sum(F,k)):
print(``):
print(`By Shalosh B. Ekhad `):
print(``):

 ope:=OperZeil(F,k,n,N)[1]:

print(`Let `):
print(``):
print(b(n)=Sum(F,k) ):
print(``):
print(`The famous Zeilberger algorithm finds (and proves!, but proof omitted) that `, b(n), `sastisfies the following linear recurrence with polynomial coefficients`):
print(``):
print(b(n)=add(coeff(ope,N,-i)*b(n-i),i=1..-ldegree(ope,N))):
print(``):
print(`and in Maple notation`):
print(``):
lprint(b(n)=add(coeff(ope,N,-i)*b(n-i),i=1..-ldegree(ope,N))):
print(``):


 d:=-ldegree(ope,N):

if K<100 then
  print(K, `must be at least 100`):
  RETURN(FAIL):
fi:

if d<2 then
print(`The recurrence has order less than 2, so we can't do anything.`):
 RETURN(FAIL):
fi:

Ini2:=[seq(add(eval(subs({n=n1,k=k1},F)),k1=0..n1),n1=0..d-1)]:

print(`Of course, the initial conditions are`):
print(seq(b(n1-1)=Ini2[n1],n1=1..d)):
print(``):

lu2:=SeqFromRec(ope,n,N,Ini2,0,2*K):

if lu2=FAIL then
 RETURN(FAIL):
fi:


 Ini1:=[0$(d-2),0,1]:

print(`Let's consider the related sequence, let's call it`, a(n) ):
print(``):
print(`satisying the same recurrence, i.e.`):
print(``):
print(a(n)=add(coeff(ope,N,-i)*a(n-i),i=1..-ldegree(ope,N))):
print(``):
print(`but with the following simpler initial conditions`):
print(``):
print(seq(a(n1-1)=Ini1[n1],n1=1..d)):

lu1:=SeqFromRec(ope,n,N,Ini1,2-d,2*K):



if lu1=FAIL then
print(`Something went wrong`):
 RETURN(FAIL):
fi:

S:=[seq(lu1[i]/lu2[i],i=1..2*K)]:

C:=S[2*K]:

print(`Using the recurrence, we can compute many terms, how about`, 2*K, `terms for each sequence and get a very good approximation to the Apery limit, i.e. the limit of`, a(n)/b(n), `as n goes to infinity`):
mu:=evalf(C):

print(`that is approximately `):
print(``):
print(mu):
print(``):

    mu1:=MyIDsLog(mu,30,10000):


      if mu1=FAIL then
         mu1:=identify(mu):
      fi:

if mu1<>mu then
print(`This constant is identified as`, mu1):
print(``):
fi:

delt:=min(seq(evalf(-log(abs(C-S[i]))/log(denom(S[i])))-1,i=K..K+10)):
print(`The implied delta is`, delt):
print(``):
if delt>0 then
print(``):
print(`Since this is positive, this suggests an Apery-style irrationality proof of`, mu1):
else
print(`Since this is negative, there is no Apery-style irrationality proof of`, mu1, `but still a very fast way to compute it to many digits`):
fi:

print(``):
print(`-----------------------`):
print(``):

print(`This took`, time()-t0, `seconds. `):

mu,delt:

end:

#DELsipur1(A,B,k,n,K): outputs an article for DEL(F,k,n,K) for all F of the form binomial(n,k)*binomial(a1*n+b1*k,k)*binomial(a2*n+b2*k,k)*x^k, for 1<=a1,a2<=A, 0<=b1,b2<=A, 1<=x<=B, try:
#DELsipur1(1,1,k,n,1000)
DELsipur1:=proc(A,B,k,n,K) local F,t0,a1,a2,b1,b2,x:
t0:=time():
print(`The Apery Limits for all summands of the form`,  binomial(n,k)*binomial(a1*n+b1*k,k)*binomial(a2*n+b2*k,k)*x^k, `for a1 and a2 from 1 to`, A, `and b1 and b2 from 0 to`, A, `and x from  to`, B):
print(``):
print(`By Shalosh B. Ekhad `):
print(``):

for a1 from 1 to A do
for a2 from 1 to A do

for b1 from 0 to A do
for b2 from 0 to A do

for x from 1 to B do
F:= binomial(n,k)*binomial(a1*n+b1*k,k)*binomial(a2*n+b2*k,k)*x^k:

DELv(F,k,n,K):
od:
od:
od:
od:
od:
print(`----------------------------------`):
print(``):
print(`This ends this book that took altogether`, time()-t0, `seconds to produce. `):



end:



#DELsipur2(A,B,k,n,K): outputs an article for DEL(F,k,n,K) for all F of the form binomial(n,k)*binomial(a1*n+b1*k,k)*binomial(a2*n+b2*k,k)*)*binomial(a3*n+b3*k,k)x^k, for 1<=a1,a2,a3<=A, 0<=b1,b2,b32<=A, 1<=x<=B, try:
#DELsipur2(1,1,k,n,1000)
DELsipur2:=proc(A,B,k,n,K) local F,t0,a1,a2,a3,b1,b2,b3,x:
t0:=time():
print(`The Apery Limits for all summands of the form`,  binomial(n,k)*binomial(a1*n+b1*k,k)*binomial(a2*n+b2*k,k)*binomial(a3*n+b3*k,k)*x^k, `for a1, a2 and a3from 1 to`, A, `and b1,b2 and b3 from 0 to`, A, `and x from  to`, B):
print(``):
print(`By Shalosh B. Ekhad `):
print(``):

for a1 from 1 to A do
for a2 from 1 to A do
for a3 from 1 to A do

for b1 from 0 to A do
for b2 from 0 to A do
for b3 from 0 to A do

for x from 1 to B do
F:= binomial(n,k)*binomial(a1*n+b1*k,k)*binomial(a2*n+b2*k,k)*binomial(a3*n+b3*k,k)*x^k:

DELv(F,k,n,K):
od:
od:
od:
od:
od:
od:
od:

print(`----------------------------------`):
print(``):
print(`This ends this book that took altogether`, time()-t0, `seconds to produce. `):



end:


#DELsipur3(n,k,K): outputs an article for DEL(F,k,n,K) for all F in the pre-computed OutF1(n,k). Try:
#DELsipur3(n,k,K)
DELsipur3:=proc(n,k,K) local gu,i,t0:
t0:=time():
gu:=OutF1(n,k):

print(`The Apery Limits for various summands that yield potential irrationality proofs`):
print(``):
print(`By Shalosh B. Ekhad `):
print(``):

for i from 1 to nops(gu) do
DELv(gu[i][1],k,n,K):
od:
print(``):
print(`-------------------------------`):
print(``):
print(`the whole thing took`, time()-t0, `seconds. `):

end:


#DELazV(P,x,K):  a verbose form of DELaz(P,x,K) (q.v.)
#DELazV(1/x+1+x,x,1000);
DELazV:=proc(P,x,K) local n,N,ope,d,Ini1,Ini2,lu1,lu2,i,n1,S,C,mu,mu1,t0,delt:

t0:=time():

print(``):
print(`-------------------------------------------------------------`):
print(``):

print(`The Apery limit generated by the sequence that are constant terms in x of`, P^n):
print(``):
print(`By Shalosh B. Ekhad `):
print(``):


 ope:=OperAZd(P^n/x,x,n,N):


 d:=-ldegree(ope,N):
if d<2 then
print(`The operator has order less than 2, so nothing an be done`):
 RETURN(FAIL):
fi:

print(`Let `, b(n), `be the coefficient of x^0 (i.e. constant term of`, P^n):
print(``):
print(`The famous Almkvist-Zeilberger algorithm finds (and proves!, but proof omitted) that `, b(n), `sastisfies the following linear recurrence with polynomial coefficients`):
print(``):
print(b(n)=add(coeff(ope,N,-i)*b(n-i),i=1..-ldegree(ope,N))):
print(``):
print(`and in Maple notation`):
print(``):
lprint(b(n)=add(coeff(ope,N,-i)*b(n-i),i=1..-ldegree(ope,N))):
print(``):





Ini2:=[seq(coeff(P^n1,x,0),n1=0..d-1)]:
lu2:=SeqFromRec(ope,n,N,Ini2,0,2*K):

if lu2=FAIL then
print(`Something went wrong`):
 RETURN(FAIL):
fi:

 Ini1:=[0$(d-2),0,1]:


print(`Let's consider the related sequence, let's call it`, a(n) ):
print(``):
print(`satisying the same recurrence, i.e.`):
print(``):
print(a(n)=add(coeff(ope,N,-i)*a(n-i),i=1..-ldegree(ope,N))):
print(``):
print(`but with the following simpler initial conditions`):
print(``):
print(seq(a(n1-1)=Ini1[n1],n1=1..d)):

lu1:=SeqFromRec(ope,n,N,Ini1,2-d,2*K):


if lu1=FAIL then
print(`Something went wrong`):
 RETURN(FAIL):
fi:

S:=[seq(lu1[i]/lu2[i],i=1..2*K)]:

C:=S[2*K]:


mu:=evalf(C):


print(`Using the recurrence, we can compute many terms, how about`, 2*K, `terms for each sequence and get a very good approximation to the Apery limit, i.e. the limit of`, a(n)/b(n), `as n goes to infinity`):

print(`that is approximately `):
print(``):
print(mu):
print(``):

    mu1:=MyIDsLog(mu,30,10000):


      if mu1=FAIL then
         mu1:=identify(mu):
      fi:

if mu1<>mu then
print(`This constant is identified as`, mu1):
print(``):
fi:

delt:=min(seq(evalf(-log(abs(C-S[i]))/log(denom(S[i])))-1,i=K..K+10)):
print(`The implied delta is`, delt):
print(``):
if delt>0 then
print(``):
print(`Since this is positive, this suggests an Apery-style irrationality proof of`, mu1):
else
print(`Since this is negative, there is no Apery-style irrationality proof of`, mu1, `but still a very fast way to compute it to many digits`):
fi:

print(``):
print(`-----------------------`):
print(``):

print(`This took`, time()-t0, `seconds. `):

mu,delt:

end:




#DELazSipur(x,K): outputs an article with the DELazV(P,x,K) for P in OutAZ1(x), giving positive deltas. Try:
#DELazSipur(x,1000):
DELazSipur:=proc(x,K) local gu,i,t0:
gu:=OutAZ1(x):

t0:=time():
for i from  1 to nops(gu) do
DELazV(gu[i][1],x,K):
od:

print(``):
print(`------------------------------`):
print(``):
print(`The whole thing took`, time()-t0, `seconds. `):
print(``):
end:




#DELazGv(F,x,n,K):  a verbose form of DELazG(F,x,n,K) (q.v.)
#DELazGv(1/x+1+x,x,1000);
DELazGv:=proc(F,x,n,K) local a,b,N,ope,d,Ini1,Ini2,lu1,lu2,i,n1,S,C,mu,mu1,t0,delt:

t0:=time():

print(``):
print(`-------------------------------------------------------------`):
print(``):

print(`The Apery limit generated by the sequences that satisfy the recurrence for the contour integral x of`, F):
print(``):
print(`By Shalosh B. Ekhad `):
print(``):


 ope:=OperAZd(F,x,n,N):

 d:=-ldegree(ope,N):
if d<2 then
print(`The operator has order less than 2, so nothing an be done`):
 RETURN(FAIL):
fi:

print(`Let `, a(n), b(n), `be the sequences satisfying the same recurrence satisfied by the contour integral around x=0 of`):
print(``):
print(F):
print(``):
print(`produced by the famous Almkvist-Zeilberger algorithm namely`):
print(``):
print(a(n)=add(coeff(ope,N,-i)*a(n-i),i=1..-ldegree(ope,N))):
print(``):
print(b(n)=add(coeff(ope,N,-i)*b(n-i),i=1..-ldegree(ope,N))):
print(``):
print(`and in Maple notation`):
print(``):
lprint(b(n)=add(coeff(ope,N,-i)*b(n-i),i=1..-ldegree(ope,N))):
print(``):
lprint(a(n)=add(coeff(ope,N,-i)*a(n-i),i=1..-ldegree(ope,N))):
print(``):

Ini2:=[1,0$(d-1)]:
Ini1:=[0,1,0$(d-2)]:

lu2:=SeqFromRec(ope,n,N,Ini2,0,2*K):

if lu2=FAIL then
print(`Something went wrong`):
 RETURN(FAIL):
fi:

 Ini1:=[0$(d-2),0,1]:


#KAKI
print(`with initial  conditions `):

print(``):
print(seq(a(n1-1)=Ini1[n1],n1=1..d)):
print(``):

print(``):
print(seq(b(n1-1)=Ini2[n1],n1=1..d)):
print(``):

lu1:=SeqFromRec(ope,n,N,Ini1,2-d,2*K):


if lu1=FAIL then
print(`Something went wrong`):
 RETURN(FAIL):
fi:

S:=[seq(lu1[i]/lu2[i],i=d+1..2*K)]:

C:=S[-1]:


mu:=evalf(C):


print(`Using the recurrence, we can compute many terms, how about`, 2*K, `terms for each sequence and get a very good approximation to the Apery limit, i.e. the limit of`, a(n)/b(n), `as n goes to infinity`):

print(`that is approximately `):
print(``):
print(mu):
print(``):

    mu1:=MyIDsLog(mu,30,10000):


      if mu1=FAIL then
         mu1:=identify(mu):
      fi:

if mu1<>mu then
print(`This constant is identified as`, mu1):
print(``):
fi:

delt:=min(seq(evalf(-log(abs(C-S[i]))/log(denom(S[i])))-1,i=K..K+10)):
print(`The implied delta is`, delt):
print(``):
if delt>0 then
print(``):
print(`Since this is positive, this suggests an Apery-style irrationality proof of`, mu1):
else
print(`Since this is negative, there is no Apery-style irrationality proof of`, mu1, `but still a very fast way to compute it to many digits`):
fi:

print(``):
print(`-----------------------`):
print(``):

print(`This took`, time()-t0, `seconds. `):

mu,delt:

end:





#CubicStory(c): An article about conjectured fast approximations to the root of 64+144*x*(1+2*a)+108*x^2*(3*a^2+3*a+1)+27*x^3*(1+2*a)=0. Try:
#CubicStory(c):
CubicStory:=proc(c) local x,ope,F,X,n,i,N,c0:
print(``):
print(`---------------------------------------`):
print(``):
print(`A conjectured good rational approximations to the cubic irrationality of`, 64+144*x*(1+2*c)+108*x^2*(3*c^2+3*c+1)+27*x^3*(1+2*c)=0, `for positive intgers c, with amazing irrationality measures as a gets bigger`):
print(``):
print(`By Shalosh B. Ekhad `):
print(``):
print(`Consider the real root of the cubic equation (with a positive)`):
print(``):
print( 64+144*x*(1+2*c)+108*x^2*(3*c^2+3*c+1)+27*x^3*(1+2*c)=0):
print(``):
print(`that according to Cardano equals`):
print(``):
print(solve(64+144*x*(1+2*c)+108*x^2*(3*c^2+3*c+1)+27*x^3*(1+2*c),x)[1]):
print(``):
print(`Consider the recurrence satisfied by the contour-integral, around x=0, of the function`):
print(``):
print(((1+c*x)*(1+(c+1)*x)/x)^n/x*x^(1/3)):
print(``):
print(`let's call it X(n)`):
print(``):
print(`that in Maple notation is`):
print(``):
F:=((1+c*x)*(1+(c+1)*x)/x)^n/x*x^(1/3):
lprint(F):
print(``):
print(`According to the famous Almkvist-Zeilberger algorithm, it satisfies the recurrence`):
print(``):
 ope:=OperAZd(F,x,n,N):
print(``):
lprint(X(n)=add(coeff(ope,N,-i)*X(n-i),i=1..-ldegree(ope,N))):
print(``):
print(`Let a(n), b(n), be the sequence that satisfy that recurrence with`):
print(``):
print(`with the initial conditions `):
print(``):
print(a(0)=0,a(1)=1):
print(``):
print(b(0)=1,b(1)=0):
print(``):
print(`We conjecture that the limit of a(n)/b(n) as n goes to infinity is the above-mentioned cubic irrationality, namely the real root of`):
print(``):
print( 64+144*x*(1+2*c)+108*x^2*(3*c^2+3*c+1)+27*x^3*(1+2*c)=0):
print(``):
print(`Not only that, these seem to give effective irrationality measures, that starting with c=4 are better than Liouville's irratioanlity measure of 3`):
print(``):
print(`The estimated irrationality measure for the case c=4 is`):
print(``):
print(1+1/DELazG( subs(c=4,((1+c*x)*(1+(c+1)*x)/x)^n/x*x^(1/3)),x,n,1000)[2]):
print(``):
print(`Let's make c bigger and see how the effective irrationaliy measure shrinks`):
print(``):

for i from 1 to 5 do
c0:=10^i+1:
print(`For c=`, c0, `the irrationality measure is estimated to be`):
print(``):
print(1+1/DELazG( subs(c=c0,((1+c*x)*(1+(c+1)*x)/x)^n/x*x^(1/3)),x,n,1000)[2]):
od:

print(``):
print(`as you can see, it seems to go down to 2`):
print(``):

end:



#DELmaz(a,b,c,K): inputs positive integers a,b,c, and outputs the Apery limit for the operaor OpeMAZ1(a,b,c,n,N) (q.v.( wuth initial conditions a(1)=1 and the rest 0, b(0)=1 and the rest 0. Try:
#DELmaz(1,1,1,1000);
DELmaz:=proc(a,b,c,K) local n,N,ope,d,Ini1,Ini2,lu1,lu2,i,S,C,mu,mu1:

ope:=OpeMAZ1(a,b,c,n,N): 
 d:=-ldegree(ope,N):
if d<2 then
 RETURN(FAIL):
fi:

Ini2:=[1,0$(d-1)]:
lu2:=SeqFromRec(ope,n,N,Ini2,0,2*K):


if lu2=FAIL then
 RETURN(FAIL):
fi:

 Ini1:=[0,1,0$(d-2)]:

lu1:=SeqFromRec(ope,n,N,Ini1,2-d,2*K):

if lu1=FAIL then
 RETURN(FAIL):
fi:

S:=[seq(lu1[i]/lu2[i],i=d+1..2*K)]:

C:=S[-1]:


if C=0 then
 RETURN(FAIL):
fi:

mu:=evalf(C):

    mu1:=MyIDsLog(mu,30,10000):
      if mu1=FAIL then
         mu:=identify(mu):
       else
       mu:=mu1:
      fi:
mu,min(seq(evalf(-log(abs(C-S[i]))/log(denom(S[i])))-1,i=K..K+10)):

end:



    

#FindMAZ(A,K): Inputs a positive  integer A and a large positive integer K
#outputs the list of pairs [[a,b,c],DELmaz(a,b,c,K)] such that DELmaz(a,b,c,K)[2] is positive
#for all 1<=a,b,c,<=A. Try
#FindMAZ(1,1000);
FindMAZ:=proc(A,K) local L,a,b,c,gu:
L:=[]:


for a from 1 to A do
 for b from 1 to A do
  for c from 1 to A do  
 if igcd(a,b,c)=1 then
      gu:=DELmaz(a,b,c,K):

       if gu<>FAIL then
          if gu[2]>0 then
          
           L:= [op(L),[[a,b,c],gu]]:

          fi:

      fi:
fi:
  od:
 od:
od:
L:
end:




#QuadStory(c): An article about conjectured fast approximations to the root of  -3*c-3/2-3*(c^2+c)^(1/2)
#QuadStory(c):
QuadStory:=proc(c) local x,ope,F,X,n,i,N,c0:
print(``):
print(`---------------------------------------`):
print(``):
print(`A conjectured good rational approximations to `,  -3*c-3/2-3*(c^2+c^(1/2)), `for positive intgers c,as good as from the continued fraction`):
print(``):
print(`By Shalosh B. Ekhad `):
print(``):
print(`Regarding the constant`,  -3*c-3/2-3*(c^2+c)^(1/2) ):
print():
print(``):
print(`Consider the recurrence satisfied by the contour-integral, around x=0, of the function`):
print(``):
print(((1+c*x)*(1+(c+1)*x)/x)^n/x*x^(1/2)):
print(``):
print(`let's call it X(n)`):
print(``):
print(`that in Maple notation is`):
print(``):
F:=((1+c*x)*(1+(c+1)*x)/x)^n/x*x^(1/2):
lprint(F):
print(``):
print(`According to the famous Almkvist-Zeilberger algorithm, it satisfies the recurrence`):
print(``):
 ope:=OperAZd(F,x,n,N):
print(``):
lprint(X(n)=add(coeff(ope,N,-i)*X(n-i),i=1..-ldegree(ope,N))):
print(``):
print(`Let a(n), b(n), be the sequence that satisfy that recurrence with`):
print(``):
print(`with the initial conditions `):
print(``):
print(a(0)=0,a(1)=1):
print(``):
print(b(0)=1,b(1)=0):
print(``):
print(`We conjecture that the limit of a(n)/b(n) as n goes to infinity is the above-mentioned`,  -3*c-3/2-3*(c^2+c)^(1/2)):
print(``):
print(`Not only that, these seem to give irrationality measure 2, just like the one coming from the continued fraction.`):
print(``):
print(`The estimated irrationality measure for the case c=4 is`):
print(``):
print(1+1/DELazG( subs(c=4,((1+c*x)*(1+(c+1)*x)/x)^n/x*x^(1/2)),x,n,1000)[2]):
print(``):
print(`Let's make c bigger and see how the effective irrationaliy measure shrinks`):
print(``):

for i from 1 to 5 do
c0:=10^i+1:
print(`For c=`, c0, `the irrationality measure is estimated to be`):
print(``):
print(1+1/DELazG( subs(c=c0,((1+c*x)*(1+(c+1)*x)/x)^n/x*x^(1/2)),x,n,1000)[2]):
od:

print(``):
print(`as you can see, it seems to go down to 2`):
print(``):

end: