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




print(`Written: Feb. 2026 `):
print(`This Version  : Feb. 2026  `):
print():
print(`This is  AvoidP.txt. a  Maple package`):
print(`accompanying  Manuel Kauers and Doron Zeilberger's  article: `):
print(` Counting Permutations avoiding  increasing subsequences proportional to their length `):

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

print(`For a list of the main procedures type: ezra(); for help with a specific procedures type ezra(ProcedureName);`):
print(`--------------------------`):
print(``):
print(`For a list of the supporting  procedures type: ezra1(); for help with a specific procedures type ezra(ProcedureName);`):
print(`--------------------------`):
print(``):
print(`For a list of the paper procedures type: ezraP(); for help with a specific procedures type ezra(ProcedureName);`):
print(`--------------------------`):
print(`For a list of the Ira Gessel procedures type: ezraI(); for help with a specific procedures type ezra(ProcedureName);`):



ezraI:=proc()
if args=NULL then

print(`The Ira Gessel  procedures are: Ira, Iv,  MatIra, `):

else
ezra(args):

fi:
end:


ezraP:=proc()
if args=NULL then

print(`The PAPER  procedures are: PaperI, PaperIc, PaperMK, PaperP, PaperPc  `):

else
ezra(args):

fi:
end:

ezra1:=proc()
if args=NULL then

print(`The SUPPORTING procedures are:  CheckIDE, fL, GD, Par, Par1, Par1A, TestMK, OpeTab, SeqFromRec, YF  `):

else
ezra(args):

fi:
end:

ezra:=proc()
if args=NULL then

print(` AviodP.txt: A Maple package for  counting permutations avoiding  increasing subsequences proportional to their length`):

print(`The MAIN procedures are:  anr,anrG,  bnr, MK, MKc, MKcF, MKg, MKgF, MKIg, MKIgC, OpeMKc, SeqarN, SeqbrN, SPseq  `):
print(``):

elif args=anr then
print(`anr(n): The number of permutations of {1,...,n} avoiding an increasing subsequence of length r. Try:`):
print(`[seq(anr(n,3),n=1..10)];`):

elif args=anrG then
print(`anrG(n,i1,i2):add(YF([op(gu1),1$i1])*YF([op(gu1), 1$i2]), gu1 in Par1A(n,r-1)). Try:`):
print(`[seq(anrG(n,3,1,1),n=1..10)];`):

elif args=bnr then
print(`bnr(n): The number of involutions of {1,...,n} avoiding an increasing subsequence of length r. Try:`):
print(`[seq(bnr(n,2),n=1..10)];`):

elif args=CheckIDE then
print(`CheckIDE(n,r): Checks that anr(n,r+1)-n*anr(n-1,r+1) equals sum of YF(p)*YF([op(p),1]) over all partitions lu of n into r parts. Try:`):
print(`CheckIDE(10,4);`):

elif args=fL then
print(`fL(L): Same as YF(L) but using the recurrence. For checking purposes only. Try:`):
print(` fL([4,3,2]) `):

elif args=Ira then
print(`Ira(m,k):The number of permutations of {1..,m} avoding an increasing subsequence of length k. Try:`):
print(`Ira(10,5);`):

elif args=Iv then
print(`Iv(v,t,N): the first N terms of I_v(t). Try; Iv(4,t,10);`):

elif args=GD then
print(`GD(L): all the children of L. Try: `):
print(` GD([4,3,2]) `):

elif args=MatIra then
print(`MatIra(J,k): The IRa Gessel determinant. Try:`):
print(`MatIra(J,4);`):

elif args=MK then
print(`MK(N): The first N terms of the sequence enumerating permutations of length 2n avoiding an increasing subsequence of length n.`):
print(`In other words: https://oeis.org/A269042. Try: `):
print(`MK(20);`):


elif args=MKc then
print(`MKc(N): The first N terms of the sequence enumerating permutations of length 2n  CONTAINING an increasing subsequence of length n.`):
print(`In other words: https://oeis.org/A269021. Try: `):
print(`MKc(20);`):


elif args=MKcF then
print(`MKcF(N): Like MKc(N), but much faster, using the Kauers-Koutschan conjectured recurrence. Try: `):
print(`MKcF(20);`):

elif args=MKg then
print(`MKg(N,r,a1,a2): The first N terms of the sequence enumerating permutations of {1,..., r*n+a1} avoiding increasing subsequences of length n+a2. The case r=2 and a1=0 and a2=0 is MK(n). Try:`):
print(`MKg(10,2,0,1);`):

elif args=MKgF then
print(`MKgCf(N,r,a1,a2): A fast version of MKgC(N,r,a1,a2) if r=2 and [a1,a2] belongs to `):
print(`OpeTab(n,Sn)[1]. Try: `):
print(`MKgF(10,2,-1,1);`):

elif args=MKgI then
print(`MKgI(N,r,a1,a2): same as MKg(N,r,a1,a2) and MKgF(N,r,a1,a2), but using Ira Gessel's determiant. Try: `):
print(`MKgI(10,2,0,1);`):

elif args=MKgC then
print(`MKgC(N,r,a1,a2): The first N terms of the sequence enumerating permutations of {1,..., r*n+a1} CONTAINING increasing subsequences of length n+a2. The case r=2 and a1=0 and a2=0 is MKc(n). Try:`):
print(`MKgC(10,2,0,1);`):

elif args=MKgCf then
print(`MKgCf(N,r,a1,a2): A fast version of MKgC(N,r,a1,a2) if r=2 and [a1,a2] belongs to `):
print(`OpeTab(n,Sn)[1]. Try: `):
print(`MKgCf(10,2,0,1);`):

elif args=MKIg then
print(`MKIg(N,r,a1,a2): The first N terms of the sequence enumerating INVOLUTIONS of {1,..., r*n+a1} avoiding increasing subsequences of length n+a2 Try:`):
print(`MKIg(10,2,0,0);`):


elif args=MKIgC then
print(`MKIgC(N,r,a1,a2): The first N terms of the sequence enumerating INVOLUTIONS of {1,..., r*n+a1} CONTAINING increasing subsequences of length n+a2 Try:`):
print(` To get https://oeis.org/A318289`):
print(`MKIgC(10,2,0,0);`):


elif args=OpeMKc then
print(`OpeMKc(n,N): The conjectured linear recurrence operator (due to M. Kauers and C. Koutschan, annihilating the sequence MKc(infinity)). Try:`):
print(`OpeMKc(n,N);`):

elif args=OpeTab then
print(`OpeTab(n,Sn): outputs the list of parameters [a1,a2] for computing MKgC(N,2,m)  followed by the annihilating operator for CONTAINING`):
print(`Try OpeTab(n,Sn);`):

elif args=PaperI then
print(`PaperI(r,M,N): inputs pos. integers r (r>=2) and M and N outputs the first N terms of the sequences enumerating  of {1, ..., r*n} AVOIDING subsequences of length n+m. Try:`):
print(`PaperI(2,2,10);`):

elif args=PaperIc then
print(`PaperIc(r,M,N): inputs pos. integers r (r>=2) and M and N outputs the first N terms of the sequences enumerating  of {1, ..., r*n} INCLUDING subsequences of length n+m. Try:`):
print(`PaperIc(2,2,10);`):

elif args=PaperMK then
print(`PaperMK(N1,N2): inputs positive integers N1 (not too big) and N2 (very big) and outputs the recurrences for the  sequences  enumerating`):
print(`permutations of {1, ..., 2*n+a1} CONTAINING an increasing subsequence of length n+a2 for [a1,a2] in`):
print(`OpeTab(n,Sn)[1]. In addition it printes out the first N1 terms (starting at n=1) and the N2-term. Try:`):
print(`PaperMK(40,200);`):

elif args=PaperP then
print(`PaperP(r,M,N): inputs pos. integers r (r>=2) and M and N outputs the first N terms of the sequences enumerating permutations of {1, ..., r*n} AVOIDING subsequences of length n+m. Try:`):
print(`PaperP(2,2,10);`):

elif args=PaperPc then
print(`PaperPc(r,M,N): inputs pos. integers r (r>=2) and M and N outputs the first N terms of the sequences enumerating permutations of {1, ..., r*n}  INCLUDING subsequences of length n+m. Try:`):
print(`PaperPc(2,2,10);`):

elif args=Par then
print(`Par(n): The set of partitions of n`):

elif args=Par1 then
print(`Par1(n,r): The set of partitions of n with r parts`):

elif args=Par1A then
print(`Par1A(n,r): The set of partitions of n at most r parts`):


elif args=SeqarN then
print(`SeqarN(r,N): [seq(anr(n,r),n=1..N)]:`):

elif args=SeqFromRec then
print(`SeqFromRec(ope,n,N,Ini,K): Given the first L-1`):
print(`terms of the sequence Ini=[f(1), ..., f(L-1)]`):
print(`satisfied by the recurrence ope(n,N)f(n)=0. Try:`):
print(`SeqFromRec(N-n-1,n,N,[1],10);`):

elif args=SeqbrN then
print(`SeqbrN(r,N): [seq(bnr(n,r),n=1..N)]:`):

elif args=SPseq then
print(`SPseq(i1,i2,N): The first n terms, starting at n=1 of sum(YF([op(p),1$i1])*YF([op(p),1$i2]), p in Par(n)), where YF(p) is the number of Standard Young tableaux of shape p. Try:`):
print(`PSeq(0,0,20) gives the first 20 terms of n! (thanks to Robinson-Schenstead). Pseq(1,0,20) gives the first 20 terms of OEIS sequence https://oeis.org/A006220. Try`):
print(`SPseq(1,0,40);`):


elif args=TestMK then
print(`TestMK(N): tests the conjectured recurrences in OpeTab(n,Sn) up to N. Try:`):
print(`TestMK(10);`):

elif args=YF then
print(`YF(L): The Young-Frobenius formula, for the number of Stansard Young tableaux of shape L. Try:`):
print(` YF([4,3,2]) `):

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

fi:


end:


with(combinat):

###start data
#OpeTab(n,Sn): outputs the list of parameters [a1,a2] for computing MKgC(N,2,m) followed by the annihilating operator.
#Try OpeTab(n,Sn);
OpeTab:=proc(n,Sn) local gu,T,recs, i,i1:
option remember:

recs := {[[-1, 0], -115502781916569600*n^2 - 849280691111086080*n^3 - 
   2306399875981587456*n^4 - 2286889438582910464*n^5 + 
   1527946474794740480*n^6 + 6015606887564800000*n^7 + 
   5560717434298071040*n^8 + 925024073884057600*n^9 - 
   2281107117825008384*n^10 - 2024839265598386688*n^11 - 
   603279317183413248*n^12 + 109843166581667840*n^13 + 
   155186662059782144*n^14 + 58584701395509248*n^15 + 
   11680138327015424*n^16 + 1248907471519744*n^17 + 56725032140800*n^18 + 
   129441370273574400*Sn + 1032076297777086720*n*Sn + 
   1940413765874158464*n^2*Sn - 942656668992991808*n^3*Sn - 
   7116831318969130432*n^4*Sn - 8106555386474854400*n^5*Sn - 
   830657713694314816*n^6*Sn + 6123760223805822080*n^7*Sn + 
   6172818605500770880*n^8*Sn + 2445450907739047168*n^9*Sn - 
   55464069879749440*n^10*Sn - 501840987104043584*n^11*Sn - 
   236855686816143744*n^12*Sn - 51331022112393728*n^13*Sn - 
   3151535974647808*n^14*Sn + 1067913835052032*n^15*Sn + 
   285478607292416*n^16*Sn + 28721527275520*n^17*Sn + 1105078190080*n^18*Sn - 
   181740716137958400*Sn^2 - 711817211042603520*n*Sn^2 - 
   73449630397126784*n^2*Sn^2 + 1953274032816437312*n^3*Sn^2 + 
   2120579141204482080*n^4*Sn^2 - 230570465000842416*n^5*Sn^2 - 
   1788730259426109968*n^6*Sn^2 - 1240860967623709168*n^7*Sn^2 - 
   222374823326070848*n^8*Sn^2 + 151101145356021104*n^9*Sn^2 + 
   104490405696128368*n^10*Sn^2 + 25563852289359472*n^11*Sn^2 + 
   908140919943104*n^12*Sn^2 - 1068989803572992*n^13*Sn^2 - 
   300489992744960*n^14*Sn^2 - 38851491808000*n^15*Sn^2 - 
   2569216706560*n^16*Sn^2 - 69067386880*n^17*Sn^2 + 16101494974265472*Sn^3 + 
   59524627038491904*n*Sn^3 - 38561943353860432*n^2*Sn^3 - 
   151960596024603460*n^3*Sn^3 - 45310521445727468*n^4*Sn^3 + 
   91058794487494164*n^5*Sn^3 + 75873336721813820*n^6*Sn^3 + 
   10707157010395492*n^7*Sn^3 - 10902599014307012*n^8*Sn^3 - 
   5666126526238228*n^9*Sn^3 - 820310142533196*n^10*Sn^3 + 
   144087599330784*n^11*Sn^3 + 75080828757392*n^12*Sn^3 + 
   12552993889280*n^13*Sn^3 + 1021288940800*n^14*Sn^3 + 
   34533693440*n^15*Sn^3 - 231209713085184*Sn^4 - 1236949089348000*n*Sn^4 + 
   1260034566799200*n^2*Sn^4 + 2482181854405308*n^3*Sn^4 - 
   513693518973942*n^4*Sn^4 - 1657127373655744*n^5*Sn^4 - 
   417011858174676*n^6*Sn^4 + 228658035155820*n^7*Sn^4 + 
   123506479044946*n^8*Sn^4 + 9399578382344*n^9*Sn^4 - 
   5444423014472*n^10*Sn^4 - 1439147777520*n^11*Sn^4 - 
   132517418240*n^12*Sn^4 - 4316711680*n^13*Sn^4 + 497393328960*Sn^5 + 
   6226844978808*n*Sn^5 - 7553930499234*n^2*Sn^5 - 8984199804111*n^3*Sn^5 + 
   6221473089621*n^4*Sn^5 + 5240246117418*n^5*Sn^5 - 868684056816*n^6*Sn^5 - 
   992409611211*n^7*Sn^5 - 49587409875*n^8*Sn^5 + 52438545720*n^9*Sn^5 + 
   7284450960*n^10*Sn^5], [[-1, 1], 1483105386954368859724800*n^2 + 
   11848771359849504354493440*n^3 + 37285284836849168298696448*n^4 + 
   54349782563701743315552256*n^5 + 20203211027770237656153856*n^6 - 
   52064202571194043023742976*n^7 - 86635838011655036470570752*n^8 - 
   51016360776117571659913216*n^9 + 4120744425365250552498432*n^10 + 
   27557215932399306814366720*n^11 + 21148654565680438359489536*n^12 + 
   8935506319366211734102016*n^13 + 2358400479956645602947072*n^14 + 
   387793332773075660455936*n^15 + 36437289078927141060608*n^16 + 
   1493839221772804685824*n^17 - 8201722656923737457817600*Sn - 
   57271108791314761823431680*n*Sn - 147357852665044280054736896*n^2*Sn - 
   145720130421586996258812416*n^3*Sn + 76766299197254208699800320*n^4*Sn + 
   383237932135123100155171712*n^5*Sn + 497747069982861844930540864*n^6*Sn + 
   366880922639741150110039232*n^7*Sn + 168431172385610105563974784*n^8*Sn + 
   46029622684559880848155648*n^9*Sn + 5369580107121224312684864*n^10*Sn - 
   601292911830885718508608*n^11*Sn - 159042372089192231260672*n^12*Sn + 
   67277676068963313963520*n^13*Sn + 33252995127121665799168*n^14*Sn + 
   6048451518969620626432*n^15*Sn + 533490749897300410368*n^16*Sn + 
   19002387772482191360*n^17*Sn + 17486392156394272955923200*Sn^2 + 
   74803469296059468296565760*n*Sn^2 + 123759389918199218613910272*n^2*Sn^2 + 
   84722828945682018015771392*n^3*Sn^2 - 12802510658351078588507152*n^4*
    Sn^2 - 68832792256980383265891856*n^5*Sn^2 - 57982900269745796334924320*
    n^6*Sn^2 - 26786037945902049373965344*n^7*Sn^2 - 
   8196304968550560705596240*n^8*Sn^2 - 2144351834845273769606288*n^9*Sn^2 - 
   670464398685782082724544*n^10*Sn^2 - 218570258619700928418624*n^11*Sn^2 - 
   53703823336268914021056*n^12*Sn^2 - 8723645030197736020224*n^13*Sn^2 - 
   880778471261656670976*n^14*Sn^2 - 49867324450652526592*n^15*Sn^2 - 
   1187649235780136960*n^16*Sn^2 - 1636663254951630097348608*Sn^3 - 
   6111796510089962379071072*n*Sn^3 - 7633252237061841320122448*n^2*Sn^3 - 
   3161095898459588123828504*n^3*Sn^3 + 1418717710241437852458420*n^4*Sn^3 + 
   2162243617747798241979956*n^5*Sn^3 + 1081234920412978791271144*n^6*Sn^3 + 
   346433192607576597934128*n^7*Sn^3 + 115431333526281079043252*n^8*Sn^3 + 
   44444523357065346963076*n^9*Sn^3 + 13420250802506174724288*n^10*Sn^3 + 
   2610712470143200497616*n^11*Sn^3 + 308156063559390289408*n^12*Sn^3 + 
   20405749513914491136*n^13*Sn^3 + 593824617890068480*n^14*Sn^3 + 
   25173824917619916362928*Sn^4 + 97826368841348120670488*n*Sn^4 + 
   92801638423378722288324*n^2*Sn^4 + 14069426883076797340410*n^3*Sn^4 - 
   23580370033085806280772*n^4*Sn^4 - 14237481861216660370652*n^5*Sn^4 - 
   4056302307222939696124*n^6*Sn^4 - 1656025386303940307550*n^7*Sn^4 - 
   831133347652437907108*n^8*Sn^4 - 242268806481322448376*n^9*Sn^4 - 
   36334489196930178800*n^10*Sn^4 - 2634225276130102272*n^11*Sn^4 - 
   74228077236258560*n^12*Sn^4 - 66104784929694784416*Sn^5 - 
   347766921054986331000*n*Sn^5 - 234433333233204707868*n^2*Sn^5 + 
   37604220869999920542*n^3*Sn^5 + 67255586484241955739*n^4*Sn^5 + 
   10598884972320015381*n^5*Sn^5 + 3620080815054389241*n^6*Sn^5 + 
   4499891715187471557*n^7*Sn^5 + 1376388085232982744*n^8*Sn^5 + 
   125259880336186320*n^9*Sn^5],
[[-1, 2], -9191577600*n^2 - 92009364480*n^3 - 397497793536*n^4 - 
   971970502144*n^5 - 1490668612992*n^6 - 1557083381184*n^7 - 
   1378816198528*n^8 - 1548717380288*n^9 - 2054481747584*n^10 - 
   2220374386240*n^11 - 1697158279296*n^12 - 868813890624*n^13 - 
   257505221632*n^14 - 4945895168*n^15 + 35839206400*n^16 + 
   19123770368*n^17 + 5630091264*n^18 + 1058844672*n^19 + 126402560*n^20 + 
   8716288*n^21 + 262144*n^22 - 9382003200*Sn + 108120925440*n*Sn + 
   901971871488*n^2*Sn + 2581622506560*n^3*Sn + 3621975628256*n^4*Sn + 
   2089334137024*n^5*Sn - 1036575129360*n^6*Sn - 2513952023648*n^7*Sn - 
   1190449694560*n^8*Sn + 899671772128*n^9*Sn + 1715105153344*n^10*Sn + 
   1255109006688*n^11*Sn + 529173476096*n^12*Sn + 115934568672*n^13*Sn - 
   5498819120*n^14*Sn - 13744632768*n^15*Sn - 5217320832*n^16*Sn - 
   1139193856*n^17*Sn - 159537920*n^18*Sn - 14150656*n^19*Sn - 
   724992*n^20*Sn - 16384*n^21*Sn - 607449600*Sn^2 - 137573134080*n*Sn^2 - 
   651738365056*n^2*Sn^2 - 1267569477184*n^3*Sn^2 - 1194327078496*n^4*Sn^2 - 
   350252377744*n^5*Sn^2 + 366002011688*n^6*Sn^2 + 381123802740*n^7*Sn^2 + 
   41859760752*n^8*Sn^2 - 152818137652*n^9*Sn^2 - 131243200792*n^10*Sn^2 - 
   52625407460*n^11*Sn^2 - 9412541600*n^12*Sn^2 + 1122335140*n^13*Sn^2 + 
   1142719168*n^14*Sn^2 + 328846160*n^15*Sn^2 + 54125504*n^16*Sn^2 + 
   5452032*n^17*Sn^2 + 316416*n^18*Sn^2 + 8192*n^19*Sn^2 - 147692160*Sn^3 + 
   9306832896*n*Sn^3 + 36123534840*n^2*Sn^3 + 53397631576*n^3*Sn^3 + 
   33440000690*n^4*Sn^3 - 1553259534*n^5*Sn^3 - 15166103902*n^6*Sn^3 - 
   7194464150*n^7*Sn^3 + 1819476166*n^8*Sn^3 + 3342982350*n^9*Sn^3 + 
   1497705150*n^10*Sn^3 + 248648710*n^11*Sn^3 - 40459520*n^12*Sn^3 - 
   28861000*n^13*Sn^3 - 6341968*n^14*Sn^3 - 723520*n^15*Sn^3 - 
   42752*n^16*Sn^3 - 1024*n^17*Sn^3 + 3991680*Sn^4 - 91755936*n*Sn^4 - 
   298618272*n^2*Sn^4 - 334357092*n^3*Sn^4 - 118960296*n^4*Sn^4 + 
   72960339*n^5*Sn^4 + 84379068*n^6*Sn^4 + 15941955*n^7*Sn^4 - 
   17299476*n^8*Sn^4 - 11677719*n^9*Sn^4 - 2223396*n^10*Sn^4 + 
   341205*n^11*Sn^4 + 214980*n^12*Sn^4 + 33264*n^13*Sn^4 + 1728*n^14*Sn^4], 
 [[0, 1], -63358848000 - 732583180800*n - 3741675897600*n^2 - 
   10947741369984*n^3 - 19322330993280*n^4 - 17275869112896*n^5 + 
   7274762204096*n^6 + 50699020505664*n^7 + 90495106844992*n^8 + 
   103941188389184*n^9 + 87789591679552*n^10 + 57168017906752*n^11 + 
   29305743476544*n^12 + 11925957892864*n^13 + 3854782193408*n^14 + 
   983607306240*n^15 + 195540294656*n^16 + 29616533504*n^17 + 
   3295596544*n^18 + 253411328*n^19 + 11993088*n^20 + 262144*n^21 + 
   404974080000*Sn + 3530532096000*n*Sn + 13849559884800*n^2*Sn + 
   31999201539840*n^3*Sn + 47482312408704*n^4*Sn + 45021100857536*n^5*Sn + 
   22188209726752*n^6*Sn - 5807151188160*n^7*Sn - 21962277876592*n^8*Sn - 
   22495450358288*n^9*Sn - 14870227508480*n^10*Sn - 7197743997344*n^11*Sn - 
   2660330746224*n^12*Sn - 762164823888*n^13*Sn - 169472279872*n^14*Sn - 
   28986169984*n^15*Sn - 3739032832*n^16*Sn - 351436032*n^17*Sn - 
   22678528*n^18*Sn - 897024*n^19*Sn - 16384*n^20*Sn - 93192120000*Sn^2 - 
   573998508000*n*Sn^2 - 1808048631600*n^2*Sn^2 - 3548289353160*n^3*Sn^2 - 
   4505906376848*n^4*Sn^2 - 3583098477580*n^5*Sn^2 - 1385020039508*n^6*Sn^2 + 
   452341703004*n^7*Sn^2 + 1073945412196*n^8*Sn^2 + 835864272956*n^9*Sn^2 + 
   416745682836*n^10*Sn^2 + 148827852172*n^11*Sn^2 + 39431723308*n^12*Sn^2 + 
   7812362704*n^13*Sn^2 + 1147043760*n^14*Sn^2 + 121575104*n^15*Sn^2 + 
   8826624*n^16*Sn^2 + 394240*n^17*Sn^2 + 8192*n^18*Sn^2 + 3248280000*Sn^3 + 
   13876233600*n*Sn^3 + 36352557120*n^2*Sn^3 + 65727414432*n^3*Sn^3 + 
   76515706956*n^4*Sn^3 + 52297201472*n^5*Sn^3 + 13439666138*n^6*Sn^3 - 
   10314309124*n^7*Sn^3 - 13326210984*n^8*Sn^3 - 7617588612*n^9*Sn^3 - 
   2779756046*n^10*Sn^3 - 697578272*n^11*Sn^3 - 121978088*n^12*Sn^3 - 
   14591600*n^13*Sn^3 - 1134912*n^14*Sn^3 - 51456*n^15*Sn^3 - 
   1024*n^16*Sn^3 - 20790000*Sn^4 - 58516200*n*Sn^4 - 130941540*n^2*Sn^4 - 
   242926434*n^3*Sn^4 - 275575500*n^4*Sn^4 - 161316921*n^5*Sn^4 - 
   15489927*n^6*Sn^4 + 46937094*n^7*Sn^4 + 38000148*n^8*Sn^4 + 
   15423273*n^9*Sn^4 + 3791667*n^10*Sn^4 + 569220*n^11*Sn^4 + 
   47952*n^12*Sn^4 + 1728*n^13*Sn^4], 
 [[0, 2], 116142336000 + 1432322611200*n + 8307115591680*n^2 + 
   30322596086784*n^3 + 78130572862080*n^4 + 149921021219968*n^5 + 
   219072674631232*n^6 + 244161108289856*n^7 + 203238315882944*n^8 + 
   117684217179840*n^9 + 34590542566592*n^10 - 13045754578496*n^11 - 
   24559003946048*n^12 - 17861593175872*n^13 - 8723583871744*n^14 - 
   3150357126912*n^15 - 864490694656*n^16 - 180597890048*n^17 - 
   28306927616*n^18 - 3224031232*n^19 - 251576320*n^20 - 11993088*n^21 - 
   262144*n^22 - 491054400000*Sn - 2651878828800*n*Sn - 
   8228496896640*n^2*Sn - 23583872253888*n^3*Sn - 59151424000896*n^4*Sn - 
   110287472223840*n^5*Sn - 145940009874352*n^6*Sn - 137000623237152*n^7*Sn - 
   90201482360960*n^8*Sn - 38666256033056*n^9*Sn - 6871068775520*n^10*Sn + 
   3900665992608*n^11*Sn + 4053287010304*n^12*Sn + 1981353892608*n^13*Sn + 
   656412903184*n^14*Sn + 158839735744*n^15*Sn + 28574455424*n^16*Sn + 
   3792551424*n^17*Sn + 361060608*n^18*Sn + 23313408*n^19*Sn + 
   913408*n^20*Sn + 16384*n^21*Sn + 414982310400*Sn^2 + 
   1554759601920*n*Sn^2 + 2426818251744*n^2*Sn^2 + 2751462331536*n^3*Sn^2 + 
   4436706015752*n^4*Sn^2 + 7694718941216*n^5*Sn^2 + 9357657611900*n^6*Sn^2 + 
   7455805825252*n^7*Sn^2 + 3794199403252*n^8*Sn^2 + 1029819238732*n^9*Sn^2 - 
   78788921612*n^10*Sn^2 - 218715519972*n^11*Sn^2 - 112155692924*n^12*Sn^2 - 
   34993090892*n^13*Sn^2 - 7553756560*n^14*Sn^2 - 1159807024*n^15*Sn^2 - 
   125304000*n^16*Sn^2 - 9111296*n^17*Sn^2 - 402432*n^18*Sn^2 - 
   8192*n^19*Sn^2 - 21934100160*Sn^3 - 67374103824*n*Sn^3 - 
   72147096648*n^2*Sn^3 - 34165935124*n^3*Sn^3 - 42210107132*n^4*Sn^3 - 
   101728125670*n^5*Sn^3 - 125384486088*n^6*Sn^3 - 85151200478*n^7*Sn^3 - 
   31275462364*n^8*Sn^3 - 2640613962*n^9*Sn^3 + 3449324344*n^10*Sn^3 + 
   2055194010*n^11*Sn^3 + 620919456*n^12*Sn^3 + 118571192*n^13*Sn^3 + 
   14777392*n^14*Sn^3 + 1164352*n^15*Sn^3 + 52480*n^16*Sn^3 + 
   1024*n^17*Sn^3 + 174303360*Sn^4 + 450715968*n*Sn^4 + 323571048*n^2*Sn^4 - 
   39952752*n^3*Sn^4 - 5020566*n^4*Sn^4 + 308413764*n^5*Sn^4 + 
   390498033*n^6*Sn^4 + 213820719*n^7*Sn^4 + 45132498*n^8*Sn^4 - 
   11800680*n^9*Sn^4 - 10851513*n^10*Sn^4 - 3342387*n^11*Sn^4 - 
   550212*n^12*Sn^4 - 47952*n^13*Sn^4 - 1728*n^14*Sn^4], 
 [[0, 3], -168699801600 - 1757424314880*n - 8018891099136*n^2 - 
   21235180910976*n^3 - 36927075190400*n^4 - 47265013884736*n^5 - 
   53718594342912*n^6 - 65645753358208*n^7 - 81609998947328*n^8 - 
   85784503366912*n^9 - 68723196681216*n^10 - 39924139884544*n^11 - 
   15611490643840*n^12 - 3002347365312*n^13 + 718829518336*n^14 + 
   830063869696*n^15 + 356252038144*n^16 + 97782262784*n^17 + 
   18597904384*n^18 + 2453573632*n^19 + 214810624*n^20 + 11206656*n^21 + 
   262144*n^22 + 411530803200*Sn + 6006075816960*n*Sn + 
   22169379013632*n^2*Sn + 35315700355584*n^3*Sn + 20786113664256*n^4*Sn - 
   13366014591808*n^5*Sn - 28040292437600*n^6*Sn - 9069789582560*n^7*Sn + 
   17175658381744*n^8*Sn + 25531321538976*n^9*Sn + 17688360644080*n^10*Sn + 
   7329961133920*n^11*Sn + 1633567859792*n^12*Sn - 46490907680*n^13*Sn - 
   182294605840*n^14*Sn - 73876721344*n^15*Sn - 17551500672*n^16*Sn - 
   2789433344*n^17*Sn - 300814592*n^18*Sn - 21191680*n^19*Sn - 
   880640*n^20*Sn - 16384*n^21*Sn - 34956662400*Sn^2 - 1725407737920*n*Sn^2 - 
   6025612277184*n^2*Sn^2 - 8104715915720*n^3*Sn^2 - 3810253664048*n^4*Sn^2 + 
   2048964124364*n^5*Sn^2 + 3176757937896*n^6*Sn^2 + 619945573024*n^7*Sn^2 - 
   1304441851704*n^8*Sn^2 - 1301118789512*n^9*Sn^2 - 590641210056*n^10*Sn^2 - 
   133311346408*n^11*Sn^2 + 1263972360*n^12*Sn^2 + 11339317660*n^13*Sn^2 + 
   4049760960*n^14*Sn^2 + 803898480*n^15*Sn^2 + 101627328*n^16*Sn^2 + 
   8180480*n^17*Sn^2 + 386048*n^18*Sn^2 + 8192*n^19*Sn^2 - 988416000*Sn^3 + 
   68519230848*n*Sn^3 + 209478116640*n^2*Sn^3 + 219768290304*n^3*Sn^3 + 
   53494876596*n^4*Sn^3 - 74493629108*n^5*Sn^3 - 58940274110*n^6*Sn^3 + 
   1621857654*n^7*Sn^3 + 22544123668*n^8*Sn^3 + 13242034028*n^9*Sn^3 + 
   3259539118*n^10*Sn^3 + 5460842*n^11*Sn^3 - 233995720*n^12*Sn^3 - 
   72776376*n^13*Sn^3 - 11484720*n^14*Sn^3 - 1035328*n^15*Sn^3 - 
   50432*n^16*Sn^3 - 1024*n^17*Sn^3 + 19768320*Sn^4 - 477171072*n*Sn^4 - 
   1239428232*n^2*Sn^4 - 957898314*n^3*Sn^4 + 22430640*n^4*Sn^4 + 
   421581573*n^5*Sn^4 + 178758024*n^6*Sn^4 - 57791109*n^7*Sn^4 - 
   74667564*n^8*Sn^4 - 23632173*n^9*Sn^4 - 679512*n^10*Sn^4 + 
   1443687*n^11*Sn^4 + 391956*n^12*Sn^4 + 42768*n^13*Sn^4 + 1728*n^14*Sn^4], 
 [[1, 0], 2192666194148977612800 + 11522015482067960970240*n + 
   26074784729137396635648*n^2 + 32330467961363950548480*n^3 + 
   21925005208241095007616*n^4 + 4530195771966625514688*n^5 - 
   5594465012324373924160*n^6 - 5872950467506818410112*n^7 - 
   2521313162442480242432*n^8 - 302257374498775051072*n^9 + 
   260955590084027061696*n^10 + 179242325843107373824*n^11 + 
   60411001413485936896*n^12 + 13048601936590202880*n^13 + 
   1885808531443721216*n^14 + 177738014938820608*n^15 + 
   9932997384990720*n^16 + 250599376830464*n^17 + 1060688936931541747200*Sn + 
   3473339957145498504960*n*Sn + 3911694700606637901312*n^2*Sn + 
   207086027077759004352*n^3*Sn - 3978895401075947346720*n^4*Sn - 
   4457074733962442586016*n^5*Sn - 2033076316694685952272*n^6*Sn + 
   142451852840850437264*n^7*Sn + 793161861772176126592*n^8*Sn + 
   560594477531754157056*n^9*Sn + 235298620737560421488*n^10*Sn + 
   68152650371940324816*n^11*Sn + 14196906170021833536*n^12*Sn + 
   2136066515334011776*n^13*Sn + 227395258420335872*n^14*Sn + 
   16280951750362368*n^15*Sn + 703932903611392*n^16*Sn + 
   13888927105024*n^17*Sn - 305601176777683836672*Sn^2 - 
   766983802670847345024*n*Sn^2 - 579108581447243140896*n^2*Sn^2 + 
   170358742194865671760*n^3*Sn^2 + 565820763201952553080*n^4*Sn^2 + 
   365600148677864965932*n^5*Sn^2 + 45428490529347866820*n^6*Sn^2 - 
   82475783032146968680*n^7*Sn^2 - 67383792264572857648*n^8*Sn^2 - 
   28376517776177978372*n^9*Sn^2 - 7876012915681173052*n^10*Sn^2 - 
   1532768662025614480*n^11*Sn^2 - 211732895413083376*n^12*Sn^2 - 
   20417904769192896*n^13*Sn^2 - 1309387555886592*n^14*Sn^2 - 
   50183057955840*n^15*Sn^2 - 868057944064*n^16*Sn^2 + 
   13064593119988773888*Sn^3 + 25750219468375048320*n*Sn^3 + 
   10085114652909016904*n^2*Sn^3 - 13304066638938213400*n^3*Sn^3 - 
   15375374815430689422*n^4*Sn^3 - 4571971918275138920*n^5*Sn^3 + 
   1983575921475215388*n^6*Sn^3 + 2322894743562726632*n^7*Sn^3 + 
   1026329014151103602*n^8*Sn^3 + 275032413208092296*n^9*Sn^3 + 
   48900020276337048*n^10*Sn^3 + 5844462379801136*n^11*Sn^3 + 
   454590340624384*n^12*Sn^3 + 20914000122112*n^13*Sn^3 + 
   434028972032*n^14*Sn^3 - 159712742344189440*Sn^4 - 
   253671162799815120*n*Sn^4 - 13380972924317984*n^2*Sn^4 + 
   179002335413852544*n^3*Sn^4 + 108289271914493180*n^4*Sn^4 - 
   3486161448595481*n^5*Sn^4 - 28817826351701545*n^6*Sn^4 - 
   14410584126939243*n^7*Sn^4 - 3698890663876699*n^8*Sn^4 - 
   563801085520756*n^9*Sn^4 - 51292419119904*n^10*Sn^4 - 
   2566778096448*n^11*Sn^4 - 54253621504*n^12*Sn^4 + 507576938679720*Sn^5 + 
   647119305857136*n*Sn^5 - 196528107738474*n^2*Sn^5 - 
   530079692647404*n^3*Sn^5 - 145901046284892*n^4*Sn^5 + 
   89049794334096*n^5*Sn^5 + 65747934518634*n^6*Sn^5 + 
   16865711308284*n^7*Sn^5 + 1996836887412*n^8*Sn^5 + 91552986288*n^9*Sn^5], 
 [[1, 1], 471710018149030080000 + 2459000370052700409600*n + 
   5048927067472821406080*n^2 + 4307101832682183951936*n^3 - 
   1108766211476108162752*n^4 - 6276561505635125681088*n^5 - 
   6321019398103174441920*n^6 - 2554568535548866276416*n^7 + 
   694037828957448510144*n^8 + 1583732786355210803136*n^9 + 
   1079689321455187206464*n^10 + 453982172869252590336*n^11 + 
   131542735256417612544*n^12 + 26943451753728000000*n^13 + 
   3857406263353322496*n^14 + 368869923151048704*n^15 + 
   21232025024466944*n^16 + 557547765153792*n^17 - 
   2889684134541680716800*Sn - 13259757041259109731840*n*Sn - 
   27676882372153926401408*n^2*Sn - 34679396677861439214144*n^3*Sn - 
   28687059458312278851040*n^4*Sn - 15807875463912981338768*n^5*Sn - 
   5032695486046424566368*n^6*Sn + 104138278792710854096*n^7*Sn + 
   1173786088841986772000*n^8*Sn + 775173719643399881104*n^9*Sn + 
   312105764442376392544*n^10*Sn + 89045991124179267568*n^11*Sn + 
   18645687517471629120*n^12*Sn + 2861746156427366272*n^13*Sn + 
   313997283388004352*n^14*Sn + 23336534154571520*n^15*Sn + 
   1052250626257920*n^16*Sn + 21710585331712*n^17*Sn + 
   447651774477118044288*Sn^2 + 1776410325880954756608*n*Sn^2 + 
   3148555268848839331232*n^2*Sn^2 + 3268262238083523353492*n^3*Sn^2 + 
   2121871328547443110516*n^4*Sn^2 + 779313640299002310228*n^5*Sn^2 + 
   20448643778558159572*n^6*Sn^2 - 164453267400073936820*n^7*Sn^2 - 
   112149366768197662036*n^8*Sn^2 - 43962369417318176228*n^9*Sn^2 - 
   11821943063529846020*n^10*Sn^2 - 2277042058219215376*n^11*Sn^2 - 
   315174096190123536*n^12*Sn^2 - 30670808357999168*n^13*Sn^2 - 
   1992312441709824*n^14*Sn^2 - 77441966181376*n^15*Sn^2 - 
   1356911583232*n^16*Sn^2 - 13700735573215552896*Sn^3 - 
   50139686898399898496*n*Sn^3 - 78361510704493606944*n^2*Sn^3 - 
   68339585695166085636*n^3*Sn^3 - 33369582865386949630*n^4*Sn^3 - 
   5113533543803940680*n^5*Sn^3 + 4707718295279726632*n^6*Sn^3 + 
   3984714021164862928*n^7*Sn^3 + 1616706782285469166*n^8*Sn^3 + 
   419394339165466220*n^9*Sn^3 + 73822261047854680*n^10*Sn^3 + 
   8832668288957920*n^11*Sn^3 + 691909612464896*n^12*Sn^3 + 
   32190846096384*n^13*Sn^3 + 678455791616*n^14*Sn^3 + 
   118169105371808460*Sn^4 + 417905314143465538*n*Sn^4 + 
   585459910380788354*n^2*Sn^4 + 427717147294332979*n^3*Sn^4 + 
   138959995526148699*n^4*Sn^4 - 26360907194767888*n^5*Sn^4 - 
   46218128851264948*n^6*Sn^4 - 21149950569643409*n^7*Sn^4 - 
   5373487854409349*n^8*Sn^4 - 830987413350772*n^9*Sn^4 - 
   77289003510736*n^10*Sn^4 - 3949649659840*n^11*Sn^4 - 
   84806973952*n^12*Sn^4 - 272907743253600*Sn^5 - 948786423022890*n*Sn^5 - 
   1184009018070519*n^2*Sn^5 - 708884826349647*n^3*Sn^5 - 
   95993674961196*n^4*Sn^5 + 132672395542548*n^5*Sn^5 + 
   85737893985639*n^6*Sn^5 + 22670360983293*n^7*Sn^5 + 
   2872571934564*n^8*Sn^5 + 143111768544*n^9*Sn^5], 
 [[1, 2], 8834066841600 + 68236231956480*n + 230089743138816*n^2 + 
   418962485736192*n^3 + 348744314451456*n^4 - 236146609282752*n^5 - 
   1181232568226880*n^6 - 1949440677105344*n^7 - 2107697143762496*n^8 - 
   1687391619296576*n^9 - 1048171998322624*n^10 - 516526808880448*n^11 - 
   204071932425920*n^12 - 64856957141760*n^13 - 16542474914048*n^14 - 
   3359712454656*n^15 - 535602381824*n^16 - 65491693568*n^17 - 
   5920436224*n^18 - 372097024*n^19 - 14483456*n^20 - 262144*n^21 - 
   12846972096000*Sn - 87836649465600*n*Sn - 269479038694400*n^2*Sn - 
   479253662073408*n^3*Sn - 522892501669920*n^4*Sn - 308860548386848*n^5*Sn + 
   18262871110544*n^6*Sn + 237316772039856*n^7*Sn + 268773182910416*n^8*Sn + 
   187348337007024*n^9*Sn + 94503015162224*n^10*Sn + 36359288647888*n^11*Sn + 
   10899719682320*n^12*Sn + 2563629350064*n^13*Sn + 471904132416*n^14*Sn + 
   67245593216*n^15*Sn + 7265387264*n^16*Sn + 574694144*n^17*Sn + 
   31349760*n^18*Sn + 1052672*n^19*Sn + 16384*n^20*Sn + 1425144123648*Sn^2 + 
   7812107318400*n*Sn^2 + 20763942755136*n^2*Sn^2 + 32822363717552*n^3*Sn^2 + 
   31862119560096*n^4*Sn^2 + 16862238787812*n^5*Sn^2 + 85581888244*n^6*Sn^2 - 
   7997839356748*n^7*Sn^2 - 7597975416252*n^8*Sn^2 - 4252791114116*n^9*Sn^2 - 
   1672488941812*n^10*Sn^2 - 487383353428*n^11*Sn^2 - 
   107184699732*n^12*Sn^2 - 17805701168*n^13*Sn^2 - 2206254800*n^14*Sn^2 - 
   198153920*n^15*Sn^2 - 12218624*n^16*Sn^2 - 463872*n^17*Sn^2 - 
   8192*n^18*Sn^2 - 29992459008*Sn^3 - 126211296512*n*Sn^3 - 
   298551338584*n^2*Sn^3 - 438174309824*n^3*Sn^3 - 383425454202*n^4*Sn^3 - 
   165376800452*n^5*Sn^3 + 23710618322*n^6*Sn^3 + 82926788896*n^7*Sn^3 + 
   59411464130*n^8*Sn^3 + 25441396668*n^9*Sn^3 + 7432985030*n^10*Sn^3 + 
   1535846104*n^11*Sn^3 + 224729016*n^12*Sn^3 + 22757200*n^13*Sn^3 + 
   1512768*n^14*Sn^3 + 59136*n^15*Sn^3 + 1024*n^16*Sn^3 + 130580736*Sn^4 + 
   396028896*n*Sn^4 + 872552088*n^2*Sn^4 + 1270732212*n^3*Sn^4 + 
   1024030566*n^4*Sn^4 + 327281817*n^5*Sn^4 - 148508853*n^6*Sn^4 - 
   213301398*n^7*Sn^4 - 111662628*n^8*Sn^4 - 34503651*n^9*Sn^4 - 
   6761973*n^10*Sn^4 - 826548*n^11*Sn^4 - 57456*n^12*Sn^4 - 1728*n^13*Sn^4],
 [[-2, 0], 24192000*n^2 + 138908160*n^3 + 227169024*n^4 - 167330688*n^5 - 
   1338824448*n^6 - 3270270912*n^7 - 4673163968*n^8 - 580783232*n^9 + 
   12507721984*n^10 + 26348063936*n^11 + 27345361600*n^12 + 
   14916464384*n^13 + 1506334976*n^14 - 4110821376*n^15 - 3502062592*n^16 - 
   1525944320*n^17 - 410816512*n^18 - 68599808*n^19 - 6488064*n^20 - 
   262144*n^21 - 338688*Sn - 103046400*n*Sn - 645537216*n^2*Sn - 
   1110901056*n^3*Sn + 839500416*n^4*Sn + 2510343568*n^5*Sn - 
   11916413664*n^6*Sn - 54214676736*n^7*Sn - 102878719904*n^8*Sn - 
   117185760864*n^9*Sn - 86007397792*n^10*Sn - 38554081216*n^11*Sn - 
   6319409504*n^12*Sn + 4418463824*n^13*Sn + 3987912640*n^14*Sn + 
   1674113664*n^15*Sn + 444599296*n^16*Sn + 77935872*n^17*Sn + 
   8748032*n^18*Sn + 569344*n^19*Sn + 16384*n^20*Sn + 470292480*Sn^2 + 
   3248944128*n*Sn^2 + 8976964608*n^2*Sn^2 + 12050874528*n^3*Sn^2 + 
   6735823744*n^4*Sn^2 + 182494840*n^5*Sn^2 + 4470135928*n^6*Sn^2 + 
   16197442908*n^7*Sn^2 + 21070928700*n^8*Sn^2 + 14889591544*n^9*Sn^2 + 
   5727028616*n^10*Sn^2 + 498729892*n^11*Sn^2 - 712292844*n^12*Sn^2 - 
   444328688*n^13*Sn^2 - 139596400*n^14*Sn^2 - 27281088*n^15*Sn^2 - 
   3377920*n^16*Sn^2 - 246784*n^17*Sn^2 - 8192*n^18*Sn^2 - 70398720*Sn^3 - 
   403150896*n*Sn^3 - 854683884*n^2*Sn^3 - 755392252*n^3*Sn^3 - 
   45134626*n^4*Sn^3 + 322974256*n^5*Sn^3 - 88386506*n^6*Sn^3 - 
   575157756*n^7*Sn^3 - 553514718*n^8*Sn^3 - 241623352*n^9*Sn^3 - 
   26524002*n^10*Sn^3 + 24642656*n^11*Sn^3 + 14015480*n^12*Sn^3 + 
   3532272*n^13*Sn^3 + 485952*n^14*Sn^3 + 35072*n^15*Sn^3 + 1024*n^16*Sn^3 + 
   1128960*Sn^4 + 5595744*n*Sn^4 + 9359628*n^2*Sn^4 + 4784142*n^3*Sn^4 - 
   3448062*n^4*Sn^4 - 4351233*n^5*Sn^4 + 987543*n^6*Sn^4 + 3996438*n^7*Sn^4 + 
   2593290*n^8*Sn^4 + 500217*n^9*Sn^4 - 200871*n^10*Sn^4 - 128580*n^11*Sn^4 - 
   25488*n^12*Sn^4 - 1728*n^13*Sn^4], 
 [[-2, -1], 18316800*n^2 + 140806656*n^3 + 309304704*n^4 - 391946880*n^5 - 
   3015276096*n^6 - 4667102208*n^7 + 1925251520*n^8 + 16774854016*n^9 + 
   23972427840*n^10 + 8348977408*n^11 - 21385811008*n^12 - 40826287616*n^13 - 
   38489100032*n^14 - 23704820736*n^15 - 10179947520*n^16 - 3084107776*n^17 - 
   647155712*n^18 - 89374720*n^19 - 7274496*n^20 - 262144*n^21 + 
   33882624*Sn + 340026624*n*Sn + 1227075072*n^2*Sn + 682626944*n^3*Sn - 
   9978696064*n^4*Sn - 42497578336*n^5*Sn - 92171817984*n^6*Sn - 
   125118862256*n^7*Sn - 105404644160*n^8*Sn - 39089191408*n^9*Sn + 
   27393979520*n^10*Sn + 55962811824*n^11*Sn + 47785733504*n^12*Sn + 
   26759282256*n^13*Sn + 10710018240*n^14*Sn + 3130832256*n^15*Sn + 
   664771072*n^16*Sn + 99785984*n^17*Sn + 10017792*n^18*Sn + 602112*n^19*Sn + 
   16384*n^20*Sn - 16212096*Sn^2 - 57183552*n*Sn^2 + 49006176*n^2*Sn^2 + 
   1635170016*n^3*Sn^2 + 7899897928*n^4*Sn^2 + 19476100960*n^5*Sn^2 + 
   28594487836*n^6*Sn^2 + 25419419040*n^7*Sn^2 + 11345787172*n^8*Sn^2 - 
   2138752208*n^9*Sn^2 - 7278037212*n^10*Sn^2 - 5800160336*n^11*Sn^2 - 
   2803824796*n^12*Sn^2 - 928725760*n^13*Sn^2 - 217045040*n^14*Sn^2 - 
   35533248*n^15*Sn^2 - 3915520*n^16*Sn^2 - 263168*n^17*Sn^2 - 
   8192*n^18*Sn^2 - 1158624*Sn^3 - 15287568*n*Sn^3 - 33633160*n^2*Sn^3 - 
   78310596*n^3*Sn^3 - 327600256*n^4*Sn^3 - 794342986*n^5*Sn^3 - 
   1021980722*n^6*Sn^3 - 687573844*n^7*Sn^3 - 126942848*n^8*Sn^3 + 
   181713050*n^9*Sn^3 + 184343474*n^10*Sn^3 + 88648440*n^11*Sn^3 + 
   26127608*n^12*Sn^3 + 4905968*n^13*Sn^3 + 569920*n^14*Sn^3 + 
   37120*n^15*Sn^3 + 1024*n^16*Sn^3 + 42840*Sn^4 + 345624*n*Sn^4 + 
   358362*n^2*Sn^4 - 79524*n^3*Sn^4 + 1676943*n^4*Sn^4 + 5720724*n^5*Sn^4 + 
   6840201*n^6*Sn^4 + 3296376*n^7*Sn^4 - 461415*n^8*Sn^4 - 1432524*n^9*Sn^4 - 
   792915*n^10*Sn^4 - 218292*n^11*Sn^4 - 30672*n^12*Sn^4 - 1728*n^13*Sn^4], 
 [[2, 1], -3075555822796800 - 23512018190653440*n - 84757331568895488*n^2 - 
   191644070566417920*n^3 - 304963487539505792*n^4 - 363222914326003072*n^5 - 
   336157537137168960*n^6 - 247704450725203968*n^7 - 147707511561044544*n^8 - 
   72043707523268992*n^9 - 28930278373061568*n^10 - 9594496303746304*n^11 - 
   2627882124870720*n^12 - 592590258850304*n^13 - 109301389416192*n^14 - 
   16313992895488*n^15 - 1938550451200*n^16 - 178960334848*n^17 - 
   12361224192*n^18 - 600555520*n^19 - 18284544*n^20 - 262144*n^21 + 
   824854314009600*Sn + 5578684820378880*n*Sn + 17468304061156096*n^2*Sn + 
   33807248322534272*n^3*Sn + 45482284742177856*n^4*Sn + 
   45306423195790048*n^5*Sn + 34725823928629856*n^6*Sn + 
   20995263026077520*n^7*Sn + 10178471528050016*n^8*Sn + 
   3998456097757776*n^9*Sn + 1280394040251552*n^10*Sn + 
   334938999141168*n^11*Sn + 71469433317024*n^12*Sn + 
   12374359554384*n^13*Sn + 1721718222272*n^14*Sn + 189552366976*n^15*Sn + 
   16126043136*n^16*Sn + 1021693184*n^17*Sn + 45349888*n^18*Sn + 
   1257472*n^19*Sn + 16384*n^20*Sn - 35781700980480*Sn^2 - 
   231547554647360*n*Sn^2 - 669267153743200*n^2*Sn^2 - 
   1165252159919200*n^3*Sn^2 - 1381525664854248*n^4*Sn^2 - 
   1191165734541264*n^5*Sn^2 - 777100111595748*n^6*Sn^2 - 
   393361513976864*n^7*Sn^2 - 156973879390956*n^8*Sn^2 - 
   49845163059008*n^9*Sn^2 - 12644446055324*n^10*Sn^2 - 
   2560047190256*n^11*Sn^2 - 411160411356*n^12*Sn^2 - 51738967872*n^13*Sn^2 - 
   4995206320*n^14*Sn^2 - 357470656*n^15*Sn^2 - 17876736*n^16*Sn^2 - 
   558080*n^17*Sn^2 - 8192*n^18*Sn^2 + 403819284480*Sn^3 + 
   2630045976064*n*Sn^3 + 7246470020960*n^2*Sn^3 + 11620934540972*n^3*Sn^3 + 
   12352919139204*n^4*Sn^3 + 9318931834766*n^5*Sn^3 + 
   5191541934006*n^6*Sn^3 + 2186964079052*n^7*Sn^3 + 705802559164*n^8*Sn^3 + 
   175381200290*n^9*Sn^3 + 33476886978*n^10*Sn^3 + 4859811512*n^11*Sn^3 + 
   526069336*n^12*Sn^3 + 41041648*n^13*Sn^3 + 2175040*n^14*Sn^3 + 
   69888*n^15*Sn^3 + 1024*n^16*Sn^3 - 1179235200*Sn^4 - 7835157840*n*Sn^4 - 
   20703785190*n^2*Sn^4 - 30706328556*n^3*Sn^4 - 29297213805*n^4*Sn^4 - 
   19257240288*n^5*Sn^4 - 9046139607*n^6*Sn^4 - 3091276920*n^7*Sn^4 - 
   771205179*n^8*Sn^4 - 139066104*n^9*Sn^4 - 17646411*n^10*Sn^4 - 
   1493556*n^11*Sn^4 - 75600*n^12*Sn^4 - 1728*n^13*Sn^4], 
 [[2, 2], -391582937088000 - 3939113493504000*n - 17565551979816960*n^2 - 
   47266585125336576*n^3 - 87182553824417280*n^4 - 118088634294329984*n^5 - 
   122517800444454784*n^6 - 100082529104206784*n^7 - 65571202837253568*n^8 - 
   34883321914248064*n^9 - 15185846157475968*n^10 - 5431619793242432*n^11 - 
   1597356311139136*n^12 - 385256865707264*n^13 - 75739496470272*n^14 - 
   12011695482880*n^15 - 1512244091904*n^16 - 147511660544*n^17 - 
   10737750016*n^18 - 548323328*n^19 - 17498112*n^20 - 262144*n^21 - 
   88517971584000*Sn + 15999265497600*n*Sn + 1687987449086400*n^2*Sn + 
   6059387838543168*n^3*Sn + 11546362674090400*n^4*Sn + 
   14613698008812304*n^5*Sn + 13425887971856672*n^6*Sn + 
   9384413518765824*n^7*Sn + 5132157428846368*n^8*Sn + 
   2234159736326944*n^9*Sn + 782148763198432*n^10*Sn + 
   221298341058752*n^11*Sn + 50627151519680*n^12*Sn + 9328518503248*n^13*Sn + 
   1372387205824*n^14*Sn + 158842557056*n^15*Sn + 14131856896*n^16*Sn + 
   931718400*n^17*Sn + 42834944*n^18*Sn + 1224704*n^19*Sn + 16384*n^20*Sn + 
   12521306649600*Sn^2 + 33406133376000*n*Sn^2 - 3835326869696*n^2*Sn^2 - 
   147035651877984*n^3*Sn^2 - 316195833112208*n^4*Sn^2 - 
   380831150046648*n^5*Sn^2 - 312309200789312*n^6*Sn^2 - 
   187721525682916*n^7*Sn^2 - 85830538338644*n^8*Sn^2 - 
   30465495282936*n^9*Sn^2 - 8482671619520*n^10*Sn^2 - 
   1858541278460*n^11*Sn^2 - 319346182028*n^12*Sn^2 - 42587750960*n^13*Sn^2 - 
   4323015920*n^14*Sn^2 - 323102400*n^15*Sn^2 - 16782080*n^16*Sn^2 - 
   541696*n^17*Sn^2 - 8192*n^18*Sn^2 - 226583859600*Sn^3 - 
   647441247600*n*Sn^3 - 380475131364*n^2*Sn^3 + 1063707109364*n^3*Sn^3 + 
   2600024941958*n^4*Sn^3 + 2930346082696*n^5*Sn^3 + 2119222597134*n^6*Sn^3 + 
   1080812425508*n^7*Sn^3 + 404984927178*n^8*Sn^3 + 113591473056*n^9*Sn^3 + 
   23975637918*n^10*Sn^3 + 3787278368*n^11*Sn^3 + 440222040*n^12*Sn^3 + 
   36456688*n^13*Sn^3 + 2029632*n^14*Sn^3 + 67840*n^15*Sn^3 + 
   1024*n^16*Sn^3 + 800785440*Sn^4 + 2234858472*n*Sn^4 + 
   1548539280*n^2*Sn^4 - 2358130314*n^3*Sn^4 - 5940696468*n^4*Sn^4 - 
   6061444821*n^5*Sn^4 - 3779131413*n^6*Sn^4 - 1589961978*n^7*Sn^4 - 
   467316096*n^8*Sn^4 - 96432003*n^9*Sn^4 - 13712127*n^10*Sn^4 - 
   1279428*n^11*Sn^4 - 70416*n^12*Sn^4 - 1728*n^13*Sn^4]}:

gu:={seq(recs[i1][1],i1=1..nops(recs))}:

for  i from 1 to nops(recs) do
T[recs[i][1]]:=recs[i][2]:
od:

gu,op(T):

end:
###end data
#OpeMKc(n,N): The conjectured linear recurrence operator (due to M. Kauers and C. Koutschan, annihilating the sequence MKc(infinity)). Try:
#OpeMKc(n,N);
OpeMKc:=proc(n,N)
-64*(1 + n)^2*(2 + n)^2*(3 + n)*(1 + 2*n)^2*(3 + 2*n)^2*(5 + 2*n)^2*(549760 + 3266000*n + 7264534*n^2 + 8663374*n^3 + 6333869*n^4 + 3012795*n^5 + 952323*n^6 + 198469*n^7 + 26156*n^8 + 1968*n^9 + 64*n^10)
+ 16*(2 + n)^2*(3 + n)*(3 + 2*n)^2*(5 + 2*n)^2*(-5543040 - 487964*n + 78563984*n^2 + 229526554*n^3 + 325846005*n^4 + 284054698*n^5 + 165789363*n^6 + 67385886*n^7 + 19359535*n^8 + 3917758*n^9 + 545913*n^10 + 49788*n^11 + 2672*n^12 + 64*n^13)*N
- 4*(3 + n)*(5 + 2*n)^2*(-250467360 - 946158512*n - 562569136*n^2 + 3271711596*n^3 + 9313604242*n^4 + 12741784568*n^5 + 11118771121*n^6 + 6753094929*n^7 + 2966908118*n^8 + 959068672*n^9 + 228837227*n^10 + 39928763*n^11 + 4969164*n^12 + 419248*n^13 + 21568*n^14
+ 512*n^15)*N^2 +
2*(-1300242000 - 6360099840*n - 10812498240*n^2 - 778719870*n^3 + 27672902302*n^4 + 55047935941*n^5 + 60004107039*n^6 + 43766004538*n^7
+ 22820793074*n^8 + 8747566435*n^9 + 2488583381*n^10 + 523718876*n^11 + 80260596*n^12 + 8677944*n^13 + 624800*n^14 + 26752*n^15 + 512*n^16)*N^3
- 3*(4 + n)*(8 + 3*n)*(10 + 3*n)*(-15900 - 61278*n - 68928*n^2 + 48699*n^3 + 204716*n^4 + 233810*n^5 + 143536*n^6 + 52389*n^7 + 11324*n^8 + 1328*n^9 + 64*n^10)*N^4:
end:


#From Findrec.txt

Yafe:=proc(ope,N) local i,ope1,coe1,L: 
if ope=0 then
 RETURN(1,0):
fi:
ope1:=expand(ope):
L:=degree(ope1,N):
coe1:=coeff(ope1,N,L):
ope1:=normal(ope1/coe1):
ope1:=normal(ope1):
ope1:=
convert(
[seq(factor(coeff(ope1,N,i))*N^i,i=ldegree(ope1,N)..degree(ope1,N))],`+`):
factor(coe1),ope1:
end:

#SeqFromRec(ope,n,N,Ini,K): Given the first L-1
#terms of the sequence Ini=[f(1), ..., f(L-1)]
#satisfied by the recurrence ope(n,N)f(n)=0
#extends it to the first K values
SeqFromRec:=proc(ope,n,N,Ini,K)
local ope1,gu,L,n1,j1:
ope1:=Yafe(ope,N)[2]:
L:=degree(ope1,N):
if nops(Ini)<>L then
 ERROR(`Ini should be of length`, L):
fi:

ope1:=expand(subs(n=n-L,ope1)/N^L):

gu:=Ini:

for n1 from nops(Ini)+1 to K do
gu:=[op(gu), -add(gu[nops(gu)+1-j1]*subs(n=n1,coeff(ope1,N,-j1)),
j1=1..L)]:
od:

gu:

end:

#End From Findrec.txt

#######
#YF(L): The Young-Frobenius formula
YF:=proc(L) local i,j,n,k:
n:=convert(L,`+`):
k:=nops(L):
n!/mul((L[i]+k-i)!,i=1..k)*mul(mul((L[j]+k-j)-(L[i]+k-i),j=1..i-1),i=1..k):
end:


#GD(L): inputs a partition and outputs its children
GD:=proc(L) local i,S:
S:={}:
for i from 1 to nops(L)-1 do
 if L[i]>L[i+1] then
  S:=S union {[op(1..i-1,L),L[i]-1,op(i+1..nops(L),L)]}:
 fi:
 od:
 if L[-1]=1 then
  S:=S union {[op(1..nops(L)-1,L)]}:
 fi:
 if L[-1]>1 then
  S:=S union {[op(1..nops(L)-1,L),L[-1]-1]}:
  fi:
 S:
 end:
 

#fL(L): YF(L) using the recurrence
fL:=proc(L) local L1,S:
option remember:
if L=[] then
  RETURN(1):
  fi:

S:=GD(L):
add(fL(L1), L1 in S):

end:


Rev:=proc(L) local i: [seq(L[nops(L)+1-i],i=1..nops(L))]:end:


#anr(n): The number of permutations of {1,...,n} avoiding an increasing subsequence of length r. Try:
#[seq(anr(n,3),n=1..10)];
anr:=proc(n,r) local gu,gu1:
if r<=0 then
 RETURN(0):
fi:

if n<r then
 RETURN(n!):
fi:

gu:=Par1A(n,r-1):
add(YF(gu1)^2, gu1 in gu):
end:


#anrG(n,i1,i2):add(YF([op(gu1),1$i1])*YF([op(gu1), 1$i2]), gu1 in Par1A(n,r-1)). Try:
#[seq(anrG(n,3,1,1),n=1..10)];
anrG:=proc(n,r,i1,i2) local gu,gu1:
if r<=0 then
 RETURN(0):
fi:

if n<r then
 RETURN(n!):
fi:

gu:=Par1A(n,r-1):
add(YF([op(gu1),1$i1])*YF([op(gu1), 1$i2]), gu1 in gu):
end:


wn:=proc(n) option remember: if n=1 then 1 : elif n=2 then 2: else wn(n-1)+(n-1)*wn(n-2):fi:end:


#bnr(n): The number of invoutions of {1,...,n} avoiding an increasing subsequence of length r. Try:
#[seq(bnr(n,3),n=1..10)];
bnr:=proc(n,r) local gu,gu1:
if r<0 then
 RETURN(0):
fi:

if n<r then
 RETURN(wn(n)):
fi:

gu:=Par1A(n,r-1):
add(YF(gu1), gu1 in gu):
end:


SeqarN:=proc(r,N) local n:
[seq(anr(n,r),n=1..N)]:
end:

SeqbrN:=proc(r,N) local n:
[seq(bnr(n,r),n=1..N)]:
end:

#MK(N): The first N terms of the sequence enumerating permutations of length 2n avoiding an increasing subsequence of length n. Try:
MK:=proc(N) local n:
option remember:
[seq(anr(2*n,n),n=1..N)]:
end:


#MKc(N): The first N terms of the sequence enumerating permutations of length 2n containing an increasing subsequence of length n. Try:
MKc:=proc(N) local L,n:
L:=MK(N):
[seq((2*n)!-L[n],n=1..N)]:
end:


#MKcF(N): Same as MKc(N) but much faster, thanks to the CONJECTURED recurrence of M. Kauers and C. Koutschan. Try:
#MKcF(100);
MKcF:=proc(N) local ope,INI,Sn,n:
ope:=OpeMKc(n,Sn):
INI:=MKc(4):
SeqFromRec(ope,n,Sn,INI,N):
end:


#MKg(N,r,a1,a2): The first N terms of the sequence enumerating permutations of {1,..., r*n+a1} avoiding increasing subsequences of length n+a2. The case r=2 and a1=0 and a2=0 is MK(n). Try:
#MKg(10,2,0,1);
MKg:=proc(N,r,a1,a2) local n:
[seq(anr(r*n+a1,n+a2),n=1..N)]:
end:

#MKgC(N,r,a1,a2): The first N terms of the sequence enumerating permutations of {1,..., r*n+a1} CONTAINING increasing subsequences of length n+a2. The case r=2 and a1=0 and a2=0 is MKc(n). Try:
#MKgC(10,2,0,1);
MKgC:=proc(N,r,a1,a2) local n,L:
L:=MKg(N,r,a1,a2):
[seq((r*n+a1)!-L[n],n=1..N)]:
end:


#MKIg(N,r,a1,a2): The first N terms of the sequence enumerating INVOLUTIONS of {1,..., r*n+a1} avoiding increasing subsequences of length n+a2 Try:
#MKIg(10,2,0,0);
MKIg:=proc(N,r,a1,a2) local n:
[seq(bnr(r*n+a1,n+a2),n=1..N)]:
end:


#MKIgC(N,r,a1,a2): The first N terms of the sequence enumerating INVOLUTIONS of {1,..., r*n+a1}  CONTAINING increasing subsequences of length n+a2 Try:
#MKIgC(10,2,0,0);
MKIgC:=proc(N,r,a1,a2) local n:
[seq(wn(r*n+a1)-bnr(r*n+a1,n+a2),n=1..N)]:
end:



#PaperP(r,M,N): inputs pos. integers r (r>=2) and M and N outputs the first N terms of the sequences of permutations of {1, ..., r*n} AVOIDING subsequences of length n+m. Try:
#PaperP(2,2,10);

PaperP:=proc(r,M,N) local m,n,t0,m1:
t0:=time():
print(`The first `, N, `terms of the sequences enumerating permutations of length`, r*n, `avoiding increasing subsequences of length`, n+m, ` for m from 0 to`, M):
print(``):
print(`By Shalosh B. Ekhad `):
print(``):
for m1 from 0 to M do
print(`The first `, N, `terms, starting with n=1,  of the sequences enumerating permutations of length`, r*n, `avoiding increasing subsequences of length`, n+m1, `are `):
print(``):
lprint(MKg(N,r,0,m1)):
print(``):
od:
print(``):
print(`--------------------`):
print(``):
print(`This ends this paper that took`, time()-t0, `seconds to produce. `):
print(``):

end:


#PaperPc(r,M,N): inputs pos. integers r (r>=2) and M and N outputs the first N terms of the sequences of permutations of {1, ..., r*n} INCLUDING subsequences of length n+m. Try:
#PaperPc(2,2,10);

PaperPc:=proc(r,M,N) local m,n,t0,m1:
t0:=time():
print(`The first `, N, `terms of the sequences enumerating permutations of length`, r*n, `INCLUDING increasing subsequences of length`, n+m, ` for m from 0 to`, M):
print(``):
print(`By Shalosh B. Ekhad `):
print(``):
for m1 from 0 to M do
print(`The first `, N, `terms, starting with n=1,  of the sequences enumerating permutations of length`, r*n, ` including increasing subsequences of length`, n+m1, `are `):
print(``):
lprint(MKgC(N,r,0,m1)):
print(``):
od:
print(``):
print(`--------------------`):
print(``):
print(`This ends this paper that took`, time()-t0, `seconds to produce. `):
print(``):

end:



#PaperI(r,M,N): inputs pos. integers r (r>=2) and M and N outputs the first N terms of the sequences of involutions of {1, ..., r*n} AVOIDING subsequences of length n+m. Try:
#PaperI(2,2,10);

PaperI:=proc(r,M,N) local m,n,t0,m1:
t0:=time():
print(`The first `, N, `terms of the sequences enumerating involutions of length`, r*n, `avoiding increasing subsequences of length`, n+m, ` for m from 0 to`, M):
print(``):
print(`By Shalosh B. Ekhad `):
print(``):
for m1 from 0 to M do
print(`The first `, N, `terms, starting with n=1,  of the sequences enumerating involutions of length`, r*n, `avoiding increasing subsequences of length`, n+m1, `are `):
print(``):
lprint(MKIg(N,r,0,m1)):
print(``):
od:
print(``):
print(`--------------------`):
print(``):
print(`This ends this paper that took`, time()-t0, `seconds to produce. `):
print(``):

end:


#PaperIc(r,M,N): inputs pos. integers r (r>=2) and M and N outputs the first N terms of the sequences of involutions of {1, ..., r*n} INCLUDING subsequences of length n+m. Try:
#PaperIc(2,2,10);

PaperIc:=proc(r,M,N) local m,n,t0,m1:
t0:=time():
print(`The first `, N, `terms of the sequences enumerating involutions of length`, r*n, `including increasing subsequences of length`, n+m, ` for m from 0 to`, M):
print(``):
print(`By Shalosh B. Ekhad `):
print(``):
for m1 from 0 to M do
print(`The first `, N, `terms, starting with n=1,  of the sequences enumerating involutions of length`, r*n, `including increasing subsequences of length`, n+m1, `are `):
print(``):
lprint(MKIgC(N,r,0,m1)):
print(``):
od:
print(``):
print(`--------------------`):
print(``):
print(`This ends this paper that took`, time()-t0, `seconds to produce. `):
print(``):

end:




#MKgF(N,r,a1,a2): A fast version of MKg(N,r,a1,a2) if r=2 and [a1,a2,0] belongs to OpeTab(n,Sn)[1]. Try:
#MKgF(10,2,-1,1);
MKgF:=proc(N,r,a1,a2) local gu,n,Sn,ope,INI,L:
gu:=OpeTab(n,Sn):

if not member([a1,a2],gu[1]) then
 RETURN(MKg(N,r,a1,a2)):
fi:

L:=MKgCf(N,r,a1,a2):

[seq((2*n+a1)!-L[n],n=1..N)]:

end:



#MKgCf(N,r,a1,a2): A fast version of MKg(N,r,a1,a2) if r=2 and [a1,a2,1] belongs to OpeTab(n,Sn)[1]. Try:
#MKgCf(10,2,0,2);
MKgCf:=proc(N,r,a1,a2) local gu,n,Sn,ope,INI:
gu:=OpeTab(n,Sn):

if not member([a1,a2],gu[1]) then
 RETURN(MKgC(N,r,a1,a2)):
fi:

ope:=gu[2][[a1,a2]]:
INI:=MKgC(degree(ope,Sn),2,a1,a2):
SeqFromRec(ope,n,Sn,INI,N):


end:


#TestMK(N): tests the conjectured recurrences in OpeTab(n,Sn) up to N. Try:
#TestMK(10);
TestMK:=proc(N) local s,Sn,n,kv:

kv:=OpeTab(n,Sn)[1]:

for s in kv do

if MKg(N,2,op(s))<>MKgF(N,2,op(s)) then
  print(s , `is no good `):
   RETURN(false):
fi:

if MKgC(N,2,op(s))<>MKgCf(N,2,op(s)) then
  print(s , `is no good `):
   RETURN(false):

fi:
od:

true:

end:



#PaperMK(N1,N2): inputs positive integers N1 (not too big) and N2 (very big) and outputs the recurrences for the  sequences  enumerating
#permutations of {1, ..., 2*n+a1} CONTAINING an increasing subsequence of length n+a2 for [a1,a2] in
#OpeTab(n,Sn)[1]. In addition it printes out the first N1 terms (starting at n=1) and the N2-term. Try:
#PaperMK(40,200);
PaperMK:=proc(N1,N2) local gu,co,n,Sn,a,i,ope,INI,i1,n1,a1,a2,T,t0:
t0:=time():
gu:=OpeTab(n,Sn):
T:=gu[2]:
gu:=gu[1]:

print(`Conjectured Linear Recurrences for permutations of length`, 2*n+a1, `CONTAINING increasing subsequences of length`, n+a2, `for the `, nops(gu), `cases  [a1,a2] in:`):
print(``):
print(gu):
print(``):
print(`By Shalosh B. Ekhad `):
print(``):

for i from 1 to  nops(gu) do
co:=gu[i]:
ope:=T[co]:
INI:=MKgCf(degree(ope,Sn),2,co[1],co[2]):
print(``):
print(`-----------------------------------------------------------`):
print(``):
print(`Theorem Number:`, i ):
print(``):
print(` Let `, a[i](n), `be the number of permutations of length `, 2*n+co[1], `contatining an increasing subsequence of length`, n+co[2]):
print(``):
print(add(coeff(ope,Sn,i1)*a[i](n+i1),i1=0..degree(ope,Sn))=0):
print(``):
print(`and in Maple notation`):
print(``):
lprint(add(coeff(ope,Sn,i1)*a[i](n+i1),i1=0..degree(ope,Sn))=0):
print(``):
print(`Subject to the initial conditions `):
print(``):
lprint(seq(a[i](n1)=INI[n1],n1=1..nops(INI) )):
print(``):
print(`For the sake of the OEIS, here are the first, starting at n=1`, N1, ` terms of the sequence `):
print(``):
lprint(MKgCf(N1,2,co[1],co[2])):
print(``):
print(`Just for fun, here is the`, N2, `-th term `):
print(``):
lprint(MKgCf(N2,2,co[1],co[2])[N2]):
od:
print(``):
print(`This ends this paper that took`, time()-t0, `seconds to produce` ):
print(``):
end:



with(linalg):
#Iv(v,t,N): the first N terms of I_v(t). Try; Iv(4,t,10);
Iv:=proc(v,t,N) local j:
add(t^(2*j+v)/j!/(j+v)!,j=0..trunc((N-v)/2)+1):
end:

#MatIra(J,k): The IRa Gessel determinant. Try:
#MatIra(J,4);
MatIra:=proc(J,k) local i,j:
[seq([seq(J(abs(i-j)),j=1..k-1)],i=1..k-1)];
end:


#Ira(m,k):The number of permutations of {1..,m} avoding an increasing subsequence of length k. Try:
#Ira(10,5);
Ira:=proc(m,k) local lu,i,j,t:
if k=1 then
 RETURN(0):
fi:

lu:=det([seq([seq(Iv(abs(i-j),t,2*m+2),j=1..k-1)],i=1..k-1)]);

coeff(lu,t,2*m)*m!^2:

end:

#MKgI(N,r,a1,a2): Same as MKg(N,r,a1,a2) but using the Ira Gessel determiant. Try:
#MKgI(10,2,0,1);
MKgI:=proc(N,r,a1,a2) local n:
[seq(Ira(r*n+a1,n+a2),n=1..N)]:
end:


#Par1(n,k): The set of partitions of n with exactly k parts
Par1:=proc(n,k) local gu,mu,mu1,i1,r:
option remember:
if n<k then
 RETURN({}):
 elif n=k then
  RETURN({[1$k]}):
 else
 gu:={}:
  for r from 1 to k do
   mu:=Par1(n-k,r):
   gu:=gu union {seq([seq(mu1[i1]+1,i1=1..r),1$(k-r)],mu1 in mu)}:
  od:
RETURN(gu):
fi:
end:

#Par1A(n,k): The set of partitions of n with <=k parts
Par1A:=proc(n,k) local k1:
{seq(op(Par1(n,k1)),k1=1..k)}:
end:



#Par(n): The set of partitions of n
Par:=proc(n) local k:option remember: { seq(op(Par1(n,k)),k=1..n)}:end:



#CheckIDE(n,r): Cheks that anr(n,r+1)-n*anr(n-1,r+1) equals sum of YF(p)*YF([op(p),1]) over all partitions lu of n into r parts. Try:
#CheckIDE(10,4);
CheckIDE:=proc(n,r) local lu,lu1:
lu:=Par1(n-1,r):
n*anr(n-1,r+1)-anr(n,r+1)-add(YF(lu1)*YF([op(lu1),1]),lu1 in lu):
end:


#SPseq(i1,i2,N): The first n terms, starting at n=1 of sum(YF([op(p),1$i1])*YF([op(p),1$i2]), p in Par(n)), where YF(p) is the number of Standard Young tableaux of shape p. Try:
#SPseq(1,0,20);
SPseq:=proc(i1,i2,N) local p,n:
[seq(add(YF([op(p),1$i1])*YF([op(p),1$i2]), p in Par(n)),n=1..N)]:
end: