MathAction zeilberger

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)abbrev package ZEIL Zeilberger
++ Author: $author
++ Date Created: $defaultdate
++ License: BSD
++ References:
++ Description: proto 
++
Zeilberger() : Exports == Implementation where

  R   ==> Integer
  X   ==> Expression R
  XX  ==> (X,X)->X
  OF  ==> OutputForm

  SUP ==> SparseUnivariatePolynomial
  FPX ==> FRAC SUPX
  NNI ==> NonNegativeInteger
  SYM ==> Symbol

  SUPX ==> SUP(X)
  NIXX ==> (NNI,X,X) -> X
  EFSP ==> ElementaryFunctionStructurePackage(R,X)
  TSPR ==> TransSolvePackage R
  LEQX ==> List Equation X
  EQSR ==> Record(eqs:LEQX,unk:List X,cert:X)

  nz ==> normalize$EFSP

  Exports == with

    term : (XX,NNI,X) -> X
    termRatio : (XX,NNI,X) -> Record(p0:X,r:X,s:X)
    gosper : (SUPX,SUPX) -> Record(pa:SUPX,pb:SUPX,pc:SUPX)
    upperBound : (Record(pa:SUPX,pb:SUPX,pc:SUPX),SUPX) -> Union(Integer,"failed")
    equationsSetup : (XX,NNI,X) -> EQSR
    solveEquations: (EQSR,NNI) -> Union(LEQX,"failed")
    recEq: (XX,NNI,X) -> Union(Equation X,"failed")

  Implementation == add 

    term(F:XX,J:NNI,k:X):X ==
      n:X:='n::X
      t:List(X):=[subscript('a,[j::OF])::X*F(n+j::X,k) for j in 0..J]
      reduce(_+,t)

    termRatio(F:XX,J:NNI,k:X):Record(p0:X,r:X,s:X) ==
      n:X:='n::X
      --q:X:=nz( term(F,J,k+1::X)/term(F,J,k) )
      r12:=nz(F(n,k+1::X)/F(n,k))
      s12:=nz(F(n,k)/F(n-1::X,k))
      r1:=numerator r12
      r2:=denominator r12
      s1:=numerator s12
      s2:=denominator s12
      P1:=(j:NNI):X+->reduce("*",[eval(s1,n,n+j::X-i::X) for i in 0..j-1],1)
      P2:=(j:NNI):X+->reduce("*",[eval(s2,n,n+l::X) for l in j+1..J],1)
      p:List(X):=[subscript('a,[j::OF])::X * P1(j) * P2(j) for j in 0..J]
      p0:=nz(reduce(_+,p))
      u1:List(X):=[eval(s2,n,n+l::X) for l in 1..J]
      u2:List(X):=[eval(s2,[n=n+l::X,k=k+1::X]) for l in 1..J]
      r:= r1 * reduce("*",u1,1)
      s:= r2 * reduce("*",u2,1)
      qrs:=nz(r/s)
      [p0,numerator qrs,denominator qrs]

    gosper(r:SUPX,s:SUPX):Record(pa:SUPX,pb:SUPX,pc:SUPX) ==
      lcr:=leadingCoefficient r
      lcs:=leadingCoefficient s
      Z:=lcr/lcs
      r:=r/lcr
      s:=s/lcs
      h:X:='h::X
      single:=create()$SingletonAsOrderedSet
      Rh:X:=resultant(r,eval(s,single,single::SUPX+h::SUPX))
      Rh0:List Equation X:=solve(Rh=0,'h)$TransSolvePackage(R)
      nums:List X:=[rhs z for z in Rh0|integer?(rhs z)$IntegerRetractions(X)]
      ints:List Integer:=[retract(u) for u in nums]
      S:=[u for u in ints | u >=0 ]
      output("** NNI zeroes of the resultant: ",S::OutputForm)$OutputPackage
      N:=#S
      p:=[subscript('p,[j::OF])::X::SUPX for j in 0..N]
      q:=[subscript('q,[j::OF])::X::SUPX for j in 0..N]
      w:=[subscript('w,[j::OF])::X::SUPX for j in 1..N]
      p.1:=r
      q.1:=s
      for j in 1..N repeat
        w.j:=gcd(p.j,eval(q.j,single,single::SUPX+(S.j)::SUPX))
        p.(j+1) := numer(p.j / w.j)
        q.(j+1) := numer(q.j / eval(w.j,single,single::SUPX-(S.j)::SUPX))
      a:=Z*p.(N+1)
      b:=q.(N+1)
      P1:=(i:NNI):SUPX+->reduce("*",[eval(w.i,single,single::SUPX-j::SUPX) for j in 1..S.i],1)      
      c:=reduce("*",[P1(i) for i in 1..N],1)
      [a,b,c]

    upperBound(g:Record(pa:SUPX,pb:SUPX,pc:SUPX),p0:SUPX):Union(Integer,"failed") ==
      single:=create()$SingletonAsOrderedSet
      a:SUPX:=g.pa
      b:SUPX:=g.pb
      c:SUPX:=g.pc * p0
      lc ==> leadingCoefficient
      BA:X:=lc(eval(b,single,single::SUPX-1::SUPX)-a)
      if degree(a) ~= degree(b) or lc a ~= lc b then
        D:List Integer:=[degree(c)-max(degree(a),degree(b))]
      else
        x:X:=BA/lc(a)
        if integer?(x)$IntegerRetractions(X) then
          D:=[degree(c)-degree(a)+1,retract x]
        else
          D:=[degree(c)-degree(a)+1]
      output("** D:=",D::OutputForm)$OutputPackage
      output("** BA:=",BA::OutputForm)$OutputPackage
      output("** lca:=",lc(a)::OutputForm)$OutputPackage
      D:=[u for u in D | u>=0 ]
      empty? D => "failed"
      max(D)

    equationsSetup(F:XX,J:NNI,k:X):EQSR ==
      single:=create()$SingletonAsOrderedSet
      tr:=termRatio(F,J::NNI,k::X)
      ur:=univariate(tr.r,kernel 'k)
      us:=univariate(tr.s,kernel 'k)
      up0:=numer univariate(tr.p0,kernel 'k) --?
      pr:=retract(ur)@SUPX
      ps:=retract(us)@SUPX
      p0:=up0 --retract(up0)@SUPX
      g:=gosper(pr,ps)
      d:=upperBound(g,up0)
      d case "failed" => [[],[],0] 
      d:=d::Integer
      b:=[subscript('b,[j::OF])::X for j in 0..d]
      x0:=reduce(_+,[b.l*k^(l-1) for l in 1..d+1],0)
      x1:=reduce(_+,[b.l*(k+1)^(l-1) for l in 1..d+1],0)
      ux0:=numer univariate(x0,kernel 'k) 
      ux1:=numer univariate(x1,kernel 'k) 
      p:=g.pc * p0
      p2:=g.pa
      p3:=eval(g.pb,single,single::SUPX - 1::SUPX)
      eqx:SUPX:=p2*ux1-p3*ux0-p
      ceqx:List X:=coefficients(eqx)
      output("** Equations:=",ceqx::OutputForm)$OutputPackage
      a:=[subscript('a,[j::OF])::X for j in 0..J]
      gp:=p3/p*ux0
      gpn:X:=retract eval(numer gp,single,k::SUPX)
      gpd:X:=retract eval(denom gp,single,k::SUPX)
      G:X:=gpn/gpd*term(F,J,k)
      [[u=0$X for u in ceqx],concat(a,b),G]

    solveEquations(e:EQSR,J:NNI):Union(LEQX,"failed") ==
      empty?(e.eqs) or empty?(e.unk) => "failed"
      x:=[u for u in e.unk | u ~= subscript('b,[0::OF])::X ]
      s:=solve(e.eqs,x)$TSPR
      empty? s => "failed"
      first s

    recEq(F:XX,J:NNI,k:X):Union(Equation X,"failed") ==
      es:=equationsSetup(F,J,k)
      sol:=solveEquations(es,J)
      if sol case "failed" then return "failed"
      else sol:=sol::LEQX
      S:=operator 'S
      n:='n::X
      rec:List X:=[rhs(sol.(i+1)) * S(n+i::X) for i in 0..J]
      rr:=reduce(_+,rec,0)   
      b0:=subscript('b,[0::OF])::X      
      rr:=numerator nz(rr/b0)
      rr=0$X</spad>

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Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/7413097682120512272-25px001.spad
      using old system compiler.
   ZEIL abbreviates package Zeilberger 
------------------------------------------------------------------------
   initializing NRLIB ZEIL for Zeilberger 
   compiling into NRLIB ZEIL 
   compiling exported term : ((Expression Integer,Expression Integer) -> Expression Integer,NonNegativeInteger,Expression Integer) -> Expression Integer
Time: 0.04 SEC.

   compiling exported termRatio : ((Expression Integer,Expression Integer) -> Expression Integer,NonNegativeInteger,Expression Integer) -> Record(p0: Expression Integer,r: Expression Integer,s: Expression Integer)
Time: 0.07 SEC.

   compiling exported gosper : (SparseUnivariatePolynomial Expression Integer,SparseUnivariatePolynomial Expression Integer) -> Record(pa: SparseUnivariatePolynomial Expression Integer,pb: SparseUnivariatePolynomial Expression Integer,pc: SparseUnivariatePolynomial Expression Integer)
Time: 0.16 SEC.

   compiling exported upperBound : (Record(pa: SparseUnivariatePolynomial Expression Integer,pb: SparseUnivariatePolynomial Expression Integer,pc: SparseUnivariatePolynomial Expression Integer),SparseUnivariatePolynomial Expression Integer) -> Union(Integer,failed)
   processing macro definition lc ==> leadingCoefficient 
Time: 0.09 SEC.

   compiling exported equationsSetup : ((Expression Integer,Expression Integer) -> Expression Integer,NonNegativeInteger,Expression Integer) -> Record(eqs: List Equation Expression Integer,unk: List Expression Integer,cert: Expression Integer)
Time: 0.24 SEC.

   compiling exported solveEquations : (Record(eqs: List Equation Expression Integer,unk: List Expression Integer,cert: Expression Integer),NonNegativeInteger) -> Union(List Equation Expression Integer,failed)
Time: 0 SEC.

   compiling exported recEq : ((Expression Integer,Expression Integer) -> Expression Integer,NonNegativeInteger,Expression Integer) -> Union(Equation Expression Integer,failed)
Time: 0.02 SEC.

(time taken in buildFunctor:  0)
Time: 0 SEC.

   Warnings: 
      [1] termRatio: not known that (AlgebraicallyClosedField) is of mode (CATEGORY domain (IF (has (Integer) (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace (Integer))) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce (% %)) (SIGNATURE number? ((Boolean) %)) (IF (has (Integer) (PolynomialFactorizationExplicit)) (ATTRIBUTE (PolynomialFactorizationExplicit)) noBranch) (SIGNATURE setSimplifyDenomsFlag ((Boolean) (Boolean))) (SIGNATURE getSimplifyDenomsFlag ((Boolean)))) noBranch))
      [2] termRatio: not known that (TranscendentalFunctionCategory) is of mode (CATEGORY domain (IF (has (Integer) (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace (Integer))) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce (% %)) (SIGNATURE number? ((Boolean) %)) (IF (has (Integer) (PolynomialFactorizationExplicit)) (ATTRIBUTE (PolynomialFactorizationExplicit)) noBranch) (SIGNATURE setSimplifyDenomsFlag ((Boolean) (Boolean))) (SIGNATURE getSimplifyDenomsFlag ((Boolean)))) noBranch))
      [3] termRatio:  k is BOTH a variable and a literal
      [4] equationsSetup:  k is BOTH a variable and a literal

   Cumulative Statistics for Constructor Zeilberger
      Time: 0.61 seconds

   finalizing NRLIB ZEIL 
   Processing Zeilberger for Browser database:
--------constructor---------
--->-->Zeilberger((term ((Expression (Integer)) (Mapping (Expression (Integer)) (Expression (Integer)) (Expression (Integer))) (NonNegativeInteger) (Expression (Integer))))): Not documented!!!!
--->-->Zeilberger((termRatio ((Record (: p0 (Expression (Integer))) (: r (Expression (Integer))) (: s (Expression (Integer)))) (Mapping (Expression (Integer)) (Expression (Integer)) (Expression (Integer))) (NonNegativeInteger) (Expression (Integer))))): Not documented!!!!
--->-->Zeilberger((gosper ((Record (: pa (SparseUnivariatePolynomial (Expression (Integer)))) (: pb (SparseUnivariatePolynomial (Expression (Integer)))) (: pc (SparseUnivariatePolynomial (Expression (Integer))))) (SparseUnivariatePolynomial (Expression (Integer))) (SparseUnivariatePolynomial (Expression (Integer)))))): Not documented!!!!
--->-->Zeilberger((upperBound ((Union (Integer) failed) (Record (: pa (SparseUnivariatePolynomial (Expression (Integer)))) (: pb (SparseUnivariatePolynomial (Expression (Integer)))) (: pc (SparseUnivariatePolynomial (Expression (Integer))))) (SparseUnivariatePolynomial (Expression (Integer)))))): Not documented!!!!
--->-->Zeilberger((equationsSetup ((Record (: eqs (List (Equation (Expression (Integer))))) (: unk (List (Expression (Integer)))) (: cert (Expression (Integer)))) (Mapping (Expression (Integer)) (Expression (Integer)) (Expression (Integer))) (NonNegativeInteger) (Expression (Integer))))): Not documented!!!!
--->-->Zeilberger((solveEquations ((Union (List (Equation (Expression (Integer)))) failed) (Record (: eqs (List (Equation (Expression (Integer))))) (: unk (List (Expression (Integer)))) (: cert (Expression (Integer)))) (NonNegativeInteger)))): Not documented!!!!
--->-->Zeilberger((recEq ((Union (Equation (Expression (Integer))) failed) (Mapping (Expression (Integer)) (Expression (Integer)) (Expression (Integer))) (NonNegativeInteger) (Expression (Integer))))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/ZEIL.NRLIB/ZEIL.lsp" (written 13 JUN 2026 05:50:04 AM):

; wrote /var/aw/var/LatexWiki/ZEIL.NRLIB/ZEIL.fasl
; compilation finished in 0:00:00.420
------------------------------------------------------------------------
   Zeilberger is now explicitly exposed in frame initial 
   Zeilberger will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/ZEIL.NRLIB/ZEIL

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-- )co zeilberger

X ==> EXPR INT

Type: Void

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S := operator 'S

$\label{eq1}S$

(1)

Type: BasicOperator?

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----
J:=1

$\label{eq2}1$

(2)

Type: PositiveInteger?

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----

F11:(X,X)->X:= (n,k) +-> binomial(n,k)

$\label{eq3}\mbox{theMap (...)}$

(3)

Type: ((Expression(Integer), Expression(Integer)) -> Expression(Integer))

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S(n) = sum(F11(n,k),k)

$\label{eq4}{S \left({n}\right)}={\sum_{ \displaystyle k}{\hbox{\axiomType{BINOMIAL}\ } \left({n , \: k}\right)}}$

(4)

Type: Equation(Expression(Integer))

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-- expected: - S(n + 1) + 2 S(n) = 0
S11:=recEq(F11,1,k)

   ** NNI zeroes of the resultant:  []
   ** D:= [0]
   ** BA:= 2
   ** lca:= - 1
   ** Equations:= [- 2 b  + a , (b  - a  - a )n + b  - a  - a ]
                        0    0    0    1    0      0    1    0

$\label{eq5}{-{S \left({n + 1}\right)}+{2 \ {S \left({n}\right)}}}= 0$

(5)

Type: Union(Equation(Expression(Integer)),...)

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F12:(X,X)->X:= (n,k) +-> binomial(n,k)^2

$\label{eq6}\mbox{theMap (...)}$

(6)

Type: ((Expression(Integer), Expression(Integer)) -> Expression(Integer))

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S(n) = sum(F12(n,k),k)

$\label{eq7}{S \left({n}\right)}={\sum_{ \displaystyle k}{{\hbox{\axiomType{BINOMIAL}\ } \left({n , \: k}\right)}^{2}}}$

(7)

Type: Equation(Expression(Integer))

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-- expected:  (- n - 1)S(n + 1) + (4 n + 2)S(n) = 0
S12:=recEq(F12,1,k)

   ** NNI zeroes of the resultant:  []
   ** D:= [1]
   ** BA:= 2 n + 2
   ** lca:= 1
   ** Equations:=
                            2
     [- 2 b n - b  - a , b n  + (- 2 b  + 2 a )n - b  - 2 b  + 2 a ,
           1     1    0   1           0      0      1      0      0
                          2
      (b  + b  - a  - a )n  + (2 b  + 2 b  - 2 a  - 2 a )n + b  + b  - a  - a ]
        1    0    1    0          1      0      1      0      1    0    1    0

$\label{eq8}{{{\left(- n - 1 \right)}\ {S \left({n + 1}\right)}}+{{\left({4 \ n}+ 2 \right)}\ {S \left({n}\right)}}}= 0$

(8)

Type: Union(Equation(Expression(Integer)),...)

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F13:(X,X)->X:= (n,k) +-> (-1)^k*binomial(2*n,k)^3   -- Dixon

$\label{eq9}\mbox{theMap (...)}$

(9)

Type: ((Expression(Integer), Expression(Integer)) -> Expression(Integer))

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S(n) = sum(F13(n,k),k)

$\label{eq10}{S \left({n}\right)}={\sum_{ \displaystyle k}{{{\left(- 1 \right)}^{k}}\ {{\hbox{\axiomType{BINOMIAL}\ } \left({{2 \ n}, \: k}\right)}^{3}}}}$

(10)

Type: Equation(Expression(Integer))

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-- expected: (- n^2  - 2 n - 1)S(n + 1) + (- 27 n^2  - 27 n - 6)S(n) = 0
S13:=recEq(F13,1,k)

   ** NNI zeroes of the resultant:  []
   ** D:= [4]
   ** BA:= 6 n + 6
   ** lca:= 1
   ** Equations:=
                                 2
     [- 6 b n - 2 b  - a , 12 b n  + (- 6 b  + 12 a )n - 6 b  - 3 b  + 9 a ,
           4       4    0      4           3       0        4      3      0

                3                           2
         - 8 b n  + (24 b  + 12 b  - 60 a )n  + (36 b  + 6 b  - 6 b  - 90 a )n
              4          4       3       0           4      3      2       0
       + 
         8 b  - 3 b  - 4 b  - 33 a
            4      3      2       0
       ,

                                   3                                      2
         (- 32 b  - 8 b  + 160 a )n  + (- 24 b  + 12 b  + 12 b  + 360 a )n
                4      3        0             4       3       2        0
       + 
         (24 b  + 30 b  + 12 b  - 6 b  + 264 a )n + 17 b  + 11 b  + b  - 5 b
              4       3       2      1        0         4       3    2      1
       + 
         63 a
             0
       ,

                  4                                     3
         - 240 a n  + (- 48 b  - 24 b  - 8 b  - 720 a )n
                0            4       3      2        0
       + 
                                            2
         (- 96 b  - 36 b  + 12 b  - 792 a )n
                4       3       1        0
       + 
         (- 54 b  - 6 b  + 18 b  + 18 b  - 6 b  - 378 a )n - 6 b  + 6 b  + 10 b
                4      3       2       1      0        0        4      3       2
       + 
         6 b  - 6 b  - 66 a
            1      0       0
       ,

                5          4                                              3
         192 a n  + 720 a n  + (- 32 b  - 24 b  - 16 b  - 8 b  + 1056 a )n
              0          0            4       3       2      1         0
       + 
                                                            2
         (- 84 b  - 60 b  - 36 b  - 12 b  + 12 b  + 756 a )n
                4       3       2       1       0        0
       + 
         (- 72 b  - 48 b  - 24 b  + 24 b  + 264 a )n - 20 b  - 12 b  - 4 b
                4       3       2       0        0         4       3      2
       + 
         4 b  + 12 b  + 36 a
            1       0       0
       ,

                           6                       5                       4
         (- 64 a  - 64 a )n  + (- 288 a  - 288 a )n  + (- 528 a  - 528 a )n
                1       0              1        0              1        0
       + 
                                                                3
         (- 8 b  - 8 b  - 8 b  - 8 b  - 8 b  - 504 a  - 504 a )n
               4      3      2      1      0        1        0
       + 
                                                                     2
         (- 24 b  - 24 b  - 24 b  - 24 b  - 24 b  - 264 a  - 264 a )n
                4       3       2       1       0        1        0
       + 
         (- 24 b  - 24 b  - 24 b  - 24 b  - 24 b  - 72 a  - 72 a )n - 8 b
                4       3       2       1       0       1       0        4
       + 
         - 8 b  - 8 b  - 8 b  - 8 b  - 8 a  - 8 a
              3      2      1      0      1      0
       ]

$\label{eq11}{{{\left(-{{n}^{2}}-{2 \ n}- 1 \right)}\ {S \left({n + 1}\right)}}+{{\left(-{{27}\ {{n}^{2}}}-{{27}\ n}- 6 \right)}\ {S \left({n}\right)}}}= 0$

(11)

Type: Union(Equation(Expression(Integer)),...)

fricas

F14:(X,X)->X:= (n,k) +-> 4^(-k)*binomial(n,2*k)*binomial(2*k,k)

$\label{eq12}\mbox{theMap (...)}$

(12)

Type: ((Expression(Integer), Expression(Integer)) -> Expression(Integer))

fricas

S(n) = sum(F14(n,k),k)

$\label{eq13}{S \left({n}\right)}={\sum_{ \displaystyle k}{{{4}^{- k}}\ {\hbox{\axiomType{BINOMIAL}\ } \left({{2 \ k}, \: k}\right)}\ {\hbox{\axiomType{BINOMIAL}\ } \left({n , \:{2 \ k}}\right)}}}$

(13)

Type: Equation(Expression(Integer))

fricas

-- expected: (- n - 1)S(n + 1) + (2 n + 1)S(n) = 0
S14:=recEq(F14,1,k)

   ** NNI zeroes of the resultant:  []
   ** D:= [0]
           2 n + 1
   ** BA:= -------
              2
   ** lca:= 1
   ** Equations:=
                              2
      - 2 b n - b  + 4 a   b n  + (b  - 4 a  - 4 a )n - 4 a  - 4 a
           0     0      0   0       0      1      0        1      0
     [-------------------, ----------------------------------------]
               2                               4

$\label{eq14}{{{\left(- n - 1 \right)}\ {S \left({n + 1}\right)}}+{{\left({2 \ n}+ 1 \right)}\ {S \left({n}\right)}}}= 0$

(14)

Type: Union(Equation(Expression(Integer)),...)

fricas

--F15:(X,X)->X:= (n,k) +-> binomial(n+k,2*k)*binomial(2*k,k)*(-1)^k/(k+1) -- only 2 eq 4 unk

F16:(X,X)->X:= (n,k) +-> (-1)^k*binomial(n,k)*binomial(2*n-2*k,n+a)

$\label{eq15}\mbox{theMap (...)}$

(15)

Type: ((Expression(Integer), Expression(Integer)) -> Expression(Integer))

fricas

S(n) = sum(F16(n,k),k)

$\label{eq16}\begin{array}{@{}l} \displaystyle {S \left({n}\right)}= \ \ \displaystyle {\sum_{ \displaystyle k}{{{\left(- 1 \right)}^{k}}\ {\hbox{\axiomType{BINOMIAL}\ } \left({n , \: k}\right)}\ {\hbox{\axiomType{BINOMIAL}\ } \left({{{2 \ n}-{2 \ k}}, \:{n + a}}\right)}}}$

(16)

Type: Equation(Expression(Integer))

fricas

-- expected: (n - a + 1)S(n + 1) + (- 2 n - 2)S(n) = 0     --***verify
S16:=recEq(F16,1,k)

   ** NNI zeroes of the resultant:  []
   ** D:= [0]
   ** BA:= - a
   ** lca:= 1
   ** Equations:=
     [(4 a  + 2 a )n + (b  + 2 a )a + 4 a  + 2 a ,
          1      0       0      0        1      0

                               2
           (b  - 16 a  - 4 a )n  + (- 2 b a + b  - 24 a  - 8 a )n
             0       1      0            0     0       1      0
         + 
                       2
           (b  + 4 a )a  - b a - 8 a  - 4 a
             0      0       0       1      0
      /
         4
       ]

$\label{eq17}{{{\left(n - a + 1 \right)}\ {S \left({n + 1}\right)}}+{{\left(-{2 \ n}- 2 \right)}\ {S \left({n}\right)}}}= 0$

(17)

Type: Union(Equation(Expression(Integer)),...)

fricas

F17:(X,X)->X:= (n,k) +-> binomial(x,k)*binomial(y,n-k)

$\label{eq18}\mbox{theMap (...)}$

(18)

Type: ((Expression(Integer), Expression(Integer)) -> Expression(Integer))

fricas

S(n) = sum(F17(n,k),k)

$\label{eq19}{S \left({n}\right)}={\sum_{ \displaystyle k}{{\hbox{\axiomType{BINOMIAL}\ } \left({x , \: k}\right)}\ {\hbox{\axiomType{BINOMIAL}\ } \left({y , \:{n - k}}\right)}}}$

(19)

Type: Equation(Expression(Integer))

fricas

-- expected: (- n - 1)S(n + 1) + (y + x - n)S(n) = 0   --***verify
S17:=recEq(F17,1,k)

   ** NNI zeroes of the resultant:  []
   ** D:= [0]
   ** BA:= y + x + 1
   ** lca:= 1
   ** Equations:=
     [- b y - b x - b  - a  + a , - a y + (b n + b )x + (a  - a )n - a ]
         0     0     0    1    0     1      0     0       1    0      0

$\label{eq20}{{{\left(- n - 1 \right)}\ {S \left({n + 1}\right)}}+{{\left(y + x - n \right)}\ {S \left({n}\right)}}}= 0$

(20)

Type: Union(Equation(Expression(Integer)),...)

fricas

F18:(X,X)->X:= (n,k) +-> k*binomial(2*n+1,2*k+1)

$\label{eq21}\mbox{theMap (...)}$

(21)

Type: ((Expression(Integer), Expression(Integer)) -> Expression(Integer))

fricas

S(n) = sum(F18(n,k),k)

$\label{eq22}{S \left({n}\right)}={\sum_{ \displaystyle k}{k \ {\hbox{\axiomType{BINOMIAL}\ } \left({{{2 \ n}+ 1}, \:{{2 \ k}+ 1}}\right)}}}$

(22)

Type: Equation(Expression(Integer))

fricas

-- expected:  (- 2 n^2  + n)S(n + 1) + (8 n^2  + 4 n)S(n) = 0  --***verify
S18:=recEq(F18,1,k)

   ** NNI zeroes of the resultant:  []
   ** D:= [1]
   ** BA:= 2 n + 1
   ** lca:= 1
   ** Equations:=
                           2
                      2 b n  + (- b  - 4 b  + 8 a )n - b  - 2 b  + 6 a
                         1         1      0      0      1      0      0
     [- 2 b n - 2 a , -------------------------------------------------,
           1       0                          2

                                       2
           (2 b  + 2 b  - 4 a  - 4 a )n  + (3 b  + 3 b  - 10 a  - 6 a )n + b
               1      0      1      0          1      0       1      0      1
         + 
           2 b  - 6 a  - 2 a
              0      1      0
      /
         2
       ]

$\label{eq23}{{{\left(-{2 \ {{n}^{2}}}+ n \right)}\ {S \left({n + 1}\right)}}+{{\left({8 \ {{n}^{2}}}+{4 \ n}\right)}\ {S \left({n}\right)}}}= 0$

(23)

Type: Union(Equation(Expression(Integer)),...)

fricas

F19:(X,X)->X:= (n,k) +-> (-1)^k*binomial(n-k,k)*2^(n-2*k)

$\label{eq24}\mbox{theMap (...)}$

(24)

Type: ((Expression(Integer), Expression(Integer)) -> Expression(Integer))

fricas

S(n) = sum(F19(n,k),k)

$\label{eq25}{S \left({n}\right)}={\sum_{ \displaystyle k}{{{\left(- 1 \right)}^{k}}\ {{2}^{n -{2 \ k}}}\ {\hbox{\axiomType{BINOMIAL}\ } \left({{n - k}, \: k}\right)}}}$

(25)

Type: Equation(Expression(Integer))

fricas

-- expected:  (n + 1)S(n + 1) + (- n - 2)S(n) = 0  --***verify
S19:=recEq(F19,1,k)

   ** NNI zeroes of the resultant:  []
   ** D:= [0]
             1
   ** BA:= - -
             2
   ** lca:= 1
                                        2
                   b  + 4 a  + 4 a   b n  + (b  - 8 a  - 4 a )n - 8 a  - 4 a
                    0      1      0   0       0      1      0        1      0
   ** Equations:= [----------------, ----------------------------------------]
                           2                             4

$\label{eq26}{{{\left(n + 1 \right)}\ {S \left({n + 1}\right)}}+{{\left(- n - 2 \right)}\ {S \left({n}\right)}}}= 0$

(26)

Type: Union(Equation(Expression(Integer)),...)

fricas

J:=2

$\label{eq27}2$

(27)

Type: PositiveInteger?

fricas

F21:(X,X)->X:= (n,k) +-> binomial(2*k,k)*binomial(n,k)*(-1/2)^k -- Reed-Dawson

$\label{eq28}\mbox{theMap (...)}$

(28)

Type: ((Expression(Integer), Expression(Integer)) -> Expression(Integer))

fricas

S(n) = sum(F21(n,k),k)

$\label{eq29}\begin{array}{@{}l} \displaystyle {S \left({n}\right)}= \ \ \displaystyle {\sum_{ \displaystyle k}{{{\left(-{\frac{1}{2}}\right)}^{k}}\ {\hbox{\axiomType{BINOMIAL}\ } \left({{2 \ k}, \: k}\right)}\ {\hbox{\axiomType{BINOMIAL}\ } \left({n , \: k}\right)}}}$

(29)

Type: Equation(Expression(Integer))

fricas

-- expected: (- n - 2)S(n + 2) + (n + 1)S(n) = 0
S21:=recEq(F21,2,k)

   ** NNI zeroes of the resultant:  []
   ** D:= [0]
   ** BA:= - 1
   ** lca:= 2
   ** Equations:=
     [b  - a , (- 2 b  + a  + 2 a )n - 3 b  + a  + 3 a ,
       0    0        0    1      0        0    1      0

                          2
         (- a  - a  - a )n  + (- b  - 3 a  - 3 a  - 3 a )n - 2 b  - 2 a  - 2 a
             2    1    0          0      2      1      0        0      2      1
       + 
         - 2 a
              0
       ]

$\label{eq30}{{{\left(- n - 2 \right)}\ {S \left({n + 2}\right)}}+{{\left(n + 1 \right)}\ {S \left({n}\right)}}}= 0$

(30)

Type: Union(Equation(Expression(Integer)),...)

fricas

F22:(X,X)->X:= (n,k) +-> 2^(-n)*(-1)^k*binomial(2*n-2*k,n-k)*binomial(n-k,k)*x^(n-2*k)

$\label{eq31}\mbox{theMap (...)}$

(31)

Type: ((Expression(Integer), Expression(Integer)) -> Expression(Integer))

fricas

S(n) = sum(F22(n,k),k)

$\label{eq32}\begin{array}{@{}l} \displaystyle {S \left({n}\right)}= \ \ \displaystyle {\sum_{ \displaystyle k}{{{\left(- 1 \right)}^{k}}\ {{2}^{- n}}\ {{x}^{n -{2 \ k}}}\ {\hbox{\axiomType{BINOMIAL}\ } \left({{n - k}, \: k}\right)}\ {\hbox{\axiomType{BINOMIAL}\ } \left({{{2 \ n}-{2 \ k}}, \:{n - k}}\right)}}}$

(32)

Type: Equation(Expression(Integer))

fricas

-- Legendre polynomials
-- expected: (n + 2)S(n + 2) + (- 2 n - 3)x S(n + 1) + (n + 1)S(n) = 0
S22:=recEq(F22,2,k)

   ** NNI zeroes of the resultant:  []
   ** D:= [0]
            2
           x  - 1
   ** BA:= ------
              2
             x
             1
   ** lca:= --
             2
            x
   ** Equations:=
             4        3                 2
      - 4 a x  - 4 a x  + (- b  - 4 a )x  + b
           2        1         0      0       0
     [----------------------------------------,
                          2
                         x

                            4                    3
           (16 a n + 16 a )x  + (12 a n + 12 a )x
                2        2           1        1
         + 
                                         2
           ((2 b  + 8 a )n + b  + 12 a )x  - 2 b n - 3 b
                0      0      0       0         0       0
      /
            2
         2 x
       ,

                    2                   4           2                  3
           (- 16 a n  - 32 a n - 12 a )x  + (- 8 a n  - 20 a n - 8 a )x
                  2         2        2            1         1       1
         + 
                   2                  2      2
           (- 4 a n  - 12 a n - 8 a )x  + b n  + 3 b n + 2 b
                 0         0       0       0        0       0
      /
            2
         4 x
       ]

$\label{eq33}{{{\left(n + 2 \right)}\ {S \left({n + 2}\right)}}+{{\left(-{2 \ n}- 3 \right)}\ x \ {S \left({n + 1}\right)}}+{{\left(n + 1 \right)}\ {S \left({n}\right)}}}= 0$

(33)

Type: Union(Equation(Expression(Integer)),...)

fricas

F23:(X,X)->X:= (n,k) +-> 2^(-k)*factorial(n)/factorial(k)/factorial(n-2*k)

$\label{eq34}\mbox{theMap (...)}$

(34)

Type: ((Expression(Integer), Expression(Integer)) -> Expression(Integer))

fricas

S(n) = sum(F23(n,k),k)

$\label{eq35}{S \left({n}\right)}={\sum_{ \displaystyle k}{\frac{{n !}\ {{2}^{- k}}}{{k !}\ {{\left(n -{2 \ k}\right)}!}}}}$

(35)

Type: Equation(Expression(Integer))

fricas

-- expected: - S(n + 2) + S(n + 1) + (n + 1)S(n) = 0 --***verify
S23:=recEq(F23,2,k)

   ** NNI zeroes of the resultant:  []
   ** D:= [0]
   ** BA:= - 2
   ** lca:= 2
   ** Equations:=
     [2 b  - 4 a , (- 2 b  + 2 a  + 4 a )n - 4 b  + 2 a  + 6 a ,
         0      0        0      1      0        0      1      0

                                     2
           (b  - 2 a  - 2 a  - 2 a )n  + (3 b  - 6 a  - 6 a  - 6 a )n + 2 b
             0      2      1      0          0      2      1      0        0
         + 
           - 4 a  - 4 a  - 4 a
                2      1      0
      /
         2
       ]

$\label{eq36}{-{S \left({n + 2}\right)}+{S \left({n + 1}\right)}+{{\left(n + 1 \right)}\ {S \left({n}\right)}}}= 0$

(36)

Type: Union(Equation(Expression(Integer)),...)