help links subscribe changes refresh edit

	SandBox
AWAIC ←	↑	→ DependentTypeTest1

CelineMethod

Edit detail for CelineMethod revision 1 of 3

	1 2 3
Editor: pagani Time: 2026/05/31 15:04:15 GMT+0
Note:

changed:
-
\begin{spad}
)abbrev package CELINE CelineMethod
CelineMethod: Exports == Implementation where
  -- Ref: https://www2.math.upenn.edu/~wilf/AeqB.pdf
  -- R:Join(Comparable,IntegralDomain,RTR,LER) 
  R   ==> Integer  
  X   ==> Expression R
  FPR ==> Fraction Polynomial R
  
  RTR ==> RetractableTo R
  LER ==> LinearlyExplicitOver R
  
  POLR ==> Polynomial R
  NNI  ==> NonNegativeInteger
  PI   ==> PositiveInteger
  BOP  ==> BasicOperator
  
  EFSP ==> ElementaryFunctionStructurePackage(Integer,X)
  UPR  ==> Union(Polynomial R,"failed")
  SMP  ==> SparseMultivariatePolynomial(Integer,Kernel(X))
  SMPF ==> SparseMultivariatePolynomial(R,Kernel(X))
  FSMP ==> Factored SMP



  Exports == with
    coeffMatrix : (PI,PI) -> Matrix X
      ++ coeffMatrix(I,J) constructs a (I+1)x(J+1) matrix of new
      ++ symbols.
    funcMatrix  : ((X,X)->X,PI,PI) -> Matrix X
      ++ funcMatrix(f,I,J) constructs a (J+1)x(I+1) matrix with
      ++ the entries M(i,j):=f(n-j,k-i)
    sumAndNormalize : (Matrix X,Matrix X) -> X
      ++ 
    retractNumeratorToPolyInt : X -> UPR
      ++
    findCoeffs : (POLR,Matrix X, Symbol) -> List List Equation FPR
      ++
    celine :((X,X)->X,PI,PI) -> Union(Matrix X,"failed")
      ++ celine(f,I,J) tries to find a recurrence for 
      ++ F(n) = summation(f(n,k),k). The returned value is either
      ++ "failed" or the coefficient matrix for the recurrence.
      ++ If successfull, then it holds that the trace of 
      ++ celine(f,I,J)*funcMatrix(f,I,J) is zero (normalize). 
    firstOrderRecurrence : ((X,X)->X,X->Matrix X) -> X 
      ++
    recRel : ((X,X)->X,PI,PI) -> Union(X,"failed")  
      ++
    recEq : ((X,X)->X,PI,PI) -> Union(Equation X,"failed")          
      ++ 


  Implementation == add

    coeffMatrix(I:PI,J:PI):Matrix X ==
      matrix [[new()$Symbol::X for j in 0..J] for i in 0..I]

    funcMatrix(F:(X,X)->X,I:PI,J:PI):Matrix X ==
      n:X:='n::X
      k:X:='k::X
      matrix [[F(n-j::X,k-i::X)/F(n,k)  for i in 0..I] for j in 0..J]

    sumAndNormalize(CM:Matrix X, FM:Matrix X):X ==
      n:=nrows(CM)
      t:X:=trace(squareMatrix(CM*FM)$SquareMatrix(n,X))
      normalize(t)$ElementaryFunctionStructurePackage(Integer,X)

    retractNumeratorToPolyInt(san:X):UPR ==
      --p:Polynomial R:=retract(numerator san)
      p:=retractIfCan(numerator san)@UPR
      p case "failed" => "failed"
      return p
 

    findCoeffs(p:POLR,m:Matrix X,k:Symbol):List List Equation FPR ==
      d:=degree(p,k)
      eqs:List Equation FPR:=[coefficient(p,k,l)::FPR=0 for l in 0..d]
      v:=members m
      x:=variables v
      sol:=solve(eqs,x)$SystemSolvePackage(R) --$TransSolvePackage(Integer)
      --first sol -- if more?
      
      
    convToFPR(M:Matrix X):Matrix FPR ==
      m:=copy(M)
      nr:=nrows(m)
      nc:=ncols(m)
      r:=zero(nr,nc)$Matrix(FPR)
      for i in 1..nr repeat
        for j in 1..nc repeat
          r(i,j):=retract m(i,j)
      return r

    convertToX(x:List Equation FPR):List Equation X ==
      [lhs(s)::X=rhs(s)::X for s in x]

    celine(F:(X,X)->X,I:PI,J:PI):Union(Matrix X,"failed") ==
      cm:Matrix X:=coeffMatrix(I,J)
      fm:=funcMatrix(F,I,J)
      san:=sumAndNormalize(cm,fm)
      p:=retractNumeratorToPolyInt(san)
      p case "failed" => "failed"
      --
      fseq:=first findCoeffs(p,cm,'k)
      seq:=convertToX(fseq)
      ev:=(x:X):X+->eval(x,seq)
      map(ev,cm)


    recRel(F:(X,X)->X,I:PI,J:PI):Union(X,"failed") ==
      C:=celine(F,I,J)
      C case "failed" => "failed"
      zero? C => "failed"
      S:=operator 'S
      SM:=funcMatrix((n,k)+->S(n),I,J)
      if C case Matrix(X) then
        M:=C*SM      
        r:=numerator reduce(_+,[M(i,i) for i in 1..I+1])
      else
        "failed"

    -- solve([r=0],[S(n)])
    -- F:=(n,k)+->binomial(n,k)*binomial(2*k,k)*(-2)^(n-k)
    
    recEq(F:(X,X)->X,I:PI,J:PI):Union(Equation X,"failed") ==
      r:Union(X,"failed"):=recRel(F,I,J)
      r case "failed" => "failed"
      S:=operator 'S
      n:='n::X
      s:=solve([r=0],[S(n)::X])$TransSolvePackage(Integer)
      empty? s => "failed"
      first first s

    firstOrderRecurrence(F:(X,X)->X,A:X->Matrix X):X ==
      n:X:='n::X
      k:X:='k::X
      --fm:=funcMatrix(F,1,1)
      a00:=(n:X):X+->elt(A(n),1,1)
      a01:=(n:X):X+->elt(A(n),1,2)
      a10:=(n:X):X+->elt(A(n),2,1)
      a11:=(n:X):X+->elt(A(n),2,2)
      anum:=(n:X):X+->a11(n)+a01(n)
      aden:=(n:X):X+->a00(n)+a10(n)
      alpha:=(n:X):X+->-anum(n)/aden(n)
      beta:=(n:X):X+->a10(n)/aden(n)*F(n+1,n+1)
      sb1:SegmentBinding X:=equation('k,segment(0,n)$Segment(X))
      sb2:SegmentBinding X:=equation('k,segment(1,n)$Segment(X))
      Q:=(n:X):X+->product(alpha(k),sb1)
      if F(0,0) ~= 0 then
        r:X:=Q(n)*(F(0,0)/alpha(0)+ summation(beta(k)/Q(k),sb2))
      else
        r:X:=Q(n)*summation(beta(k)/Q(k),sb2)
      return r
      

-- https://en.wikipedia.org/wiki/Recurrence_relation

\end{spad}

\begin{axiom}

)version

X ==> EXPR INT


F1:=(n,k)+->binomial(n,k)
r1:=recEq(F1,1,1)

-- Example 4.3.1. 
F2:=(n,k)+->binomial(n,k)^2
r2:=recEq(F2,2,2)

-- Example 4.3.3. 
F3:=(n,k)+->binomial(n,k)*binomial(2*k,k)*(-2)^(n-k)
r3:=recEq(F3,2,2)

-- Example 4.3.2. 
F4:=(n,k)+->(-1)^k*binomial(n,k)*x^k/factorial(k)
r4:=recEq(F4,2,2)  -- (1,2) already works!

-- Example 4.1.1.
F5:=(n,k)+->k*binomial(n,k)
r5:=recEq(F5,1,1)


F6:=(n,k)+->k^2*binomial(n,k)
r6:=recEq(F6,1,1) -- fails, but (1,2) and (2,1) are working .... pref: 1,2

r6_12:=recEq(F6,1,2)
r6_21:=recEq(F6,2,1)

r6_21 - r612

\end{axiom}

fricas

(1) -> <spad>

fricas

)abbrev package CELINE CelineMethod
CelineMethod: Exports == Implementation where
  -- Ref: https://www2.math.upenn.edu/~wilf/AeqB.pdf
  -- R:Join(Comparable,IntegralDomain,RTR,LER) 
  R   ==> Integer  
  X   ==> Expression R
  FPR ==> Fraction Polynomial R

  RTR ==> RetractableTo R
  LER ==> LinearlyExplicitOver R

  POLR ==> Polynomial R
  NNI  ==> NonNegativeInteger
  PI   ==> PositiveInteger
  BOP  ==> BasicOperator

  EFSP ==> ElementaryFunctionStructurePackage(Integer,X)
  UPR  ==> Union(Polynomial R,"failed")
  SMP  ==> SparseMultivariatePolynomial(Integer,Kernel(X))
  SMPF ==> SparseMultivariatePolynomial(R,Kernel(X))
  FSMP ==> Factored SMP

  Exports == with
    coeffMatrix : (PI,PI) -> Matrix X
      ++ coeffMatrix(I,J) constructs a (I+1)x(J+1) matrix of new
      ++ symbols.
    funcMatrix  : ((X,X)->X,PI,PI) -> Matrix X
      ++ funcMatrix(f,I,J) constructs a (J+1)x(I+1) matrix with
      ++ the entries M(i,j):=f(n-j,k-i)
    sumAndNormalize : (Matrix X,Matrix X) -> X
      ++ 
    retractNumeratorToPolyInt : X -> UPR
      ++
    findCoeffs : (POLR,Matrix X, Symbol) -> List List Equation FPR
      ++
    celine :((X,X)->X,PI,PI) -> Union(Matrix X,"failed")
      ++ celine(f,I,J) tries to find a recurrence for 
      ++ F(n) = summation(f(n,k),k). The returned value is either
      ++ "failed" or the coefficient matrix for the recurrence.
      ++ If successfull, then it holds that the trace of 
      ++ celine(f,I,J)*funcMatrix(f,I,J) is zero (normalize). 
    firstOrderRecurrence : ((X,X)->X,X->Matrix X) -> X 
      ++
    recRel : ((X,X)->X,PI,PI) -> Union(X,"failed")  
      ++
    recEq : ((X,X)->X,PI,PI) -> Union(Equation X,"failed")          
      ++ 

  Implementation == add

    coeffMatrix(I:PI,J:PI):Matrix X ==
      matrix [[new()$Symbol::X for j in 0..J] for i in 0..I]

    funcMatrix(F:(X,X)->X,I:PI,J:PI):Matrix X ==
      n:X:='n::X
      k:X:='k::X
      matrix [[F(n-j::X,k-i::X)/F(n,k)  for i in 0..I] for j in 0..J]

    sumAndNormalize(CM:Matrix X, FM:Matrix X):X ==
      n:=nrows(CM)
      t:X:=trace(squareMatrix(CM*FM)$SquareMatrix(n,X))
      normalize(t)$ElementaryFunctionStructurePackage(Integer,X)

    retractNumeratorToPolyInt(san:X):UPR ==
      --p:Polynomial R:=retract(numerator san)
      p:=retractIfCan(numerator san)@UPR
      p case "failed" => "failed"
      return p

    findCoeffs(p:POLR,m:Matrix X,k:Symbol):List List Equation FPR ==
      d:=degree(p,k)
      eqs:List Equation FPR:=[coefficient(p,k,l)::FPR=0 for l in 0..d]
      v:=members m
      x:=variables v
      sol:=solve(eqs,x)$SystemSolvePackage(R) --$TransSolvePackage(Integer)
      --first sol -- if more?

    convToFPR(M:Matrix X):Matrix FPR ==
      m:=copy(M)
      nr:=nrows(m)
      nc:=ncols(m)
      r:=zero(nr,nc)$Matrix(FPR)
      for i in 1..nr repeat
        for j in 1..nc repeat
          r(i,j):=retract m(i,j)
      return r

    convertToX(x:List Equation FPR):List Equation X ==
      [lhs(s)::X=rhs(s)::X for s in x]

    celine(F:(X,X)->X,I:PI,J:PI):Union(Matrix X,"failed") ==
      cm:Matrix X:=coeffMatrix(I,J)
      fm:=funcMatrix(F,I,J)
      san:=sumAndNormalize(cm,fm)
      p:=retractNumeratorToPolyInt(san)
      p case "failed" => "failed"
      --
      fseq:=first findCoeffs(p,cm,'k)
      seq:=convertToX(fseq)
      ev:=(x:X):X+->eval(x,seq)
      map(ev,cm)

    recRel(F:(X,X)->X,I:PI,J:PI):Union(X,"failed") ==
      C:=celine(F,I,J)
      C case "failed" => "failed"
      zero? C => "failed"
      S:=operator 'S
      SM:=funcMatrix((n,k)+->S(n),I,J)
      if C case Matrix(X) then
        M:=C*SM      
        r:=numerator reduce(_+,[M(i,i) for i in 1..I+1])
      else
        "failed"

    -- solve([r=0],[S(n)])
    -- F:=(n,k)+->binomial(n,k)*binomial(2*k,k)*(-2)^(n-k)

    recEq(F:(X,X)->X,I:PI,J:PI):Union(Equation X,"failed") ==
      r:Union(X,"failed"):=recRel(F,I,J)
      r case "failed" => "failed"
      S:=operator 'S
      n:='n::X
      s:=solve([r=0],[S(n)::X])$TransSolvePackage(Integer)
      empty? s => "failed"
      first first s

    firstOrderRecurrence(F:(X,X)->X,A:X->Matrix X):X ==
      n:X:='n::X
      k:X:='k::X
      --fm:=funcMatrix(F,1,1)
      a00:=(n:X):X+->elt(A(n),1,1)
      a01:=(n:X):X+->elt(A(n),1,2)
      a10:=(n:X):X+->elt(A(n),2,1)
      a11:=(n:X):X+->elt(A(n),2,2)
      anum:=(n:X):X+->a11(n)+a01(n)
      aden:=(n:X):X+->a00(n)+a10(n)
      alpha:=(n:X):X+->-anum(n)/aden(n)
      beta:=(n:X):X+->a10(n)/aden(n)*F(n+1,n+1)
      sb1:SegmentBinding X:=equation('k,segment(0,n)$Segment(X))
      sb2:SegmentBinding X:=equation('k,segment(1,n)$Segment(X))
      Q:=(n:X):X+->product(alpha(k),sb1)
      if F(0,0) ~= 0 then
        r:X:=Q(n)*(F(0,0)/alpha(0)+ summation(beta(k)/Q(k),sb2))
      else
        r:X:=Q(n)*summation(beta(k)/Q(k),sb2)
      return r

-- https://en.wikipedia.org/wiki/Recurrence_relation</spad>

fricas

Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/5910754882400853191-25px001.spad
      using old system compiler.
   CELINE abbreviates package CelineMethod 
------------------------------------------------------------------------
   initializing NRLIB CELINE for CelineMethod 
   compiling into NRLIB CELINE 
   compiling exported coeffMatrix : (PositiveInteger,PositiveInteger) -> Matrix Expression Integer
Time: 0.03 SEC.

   compiling exported funcMatrix : ((Expression Integer,Expression Integer) -> Expression Integer,PositiveInteger,PositiveInteger) -> Matrix Expression Integer
Time: 0.01 SEC.

   compiling exported sumAndNormalize : (Matrix Expression Integer,Matrix Expression Integer) -> Expression Integer
Time: 0.02 SEC.

   compiling exported retractNumeratorToPolyInt : Expression Integer -> Union(Polynomial Integer,failed)
Time: 0 SEC.

   compiling exported findCoeffs : (Polynomial Integer,Matrix Expression Integer,Symbol) -> List List Equation Fraction Polynomial Integer
Time: 0.01 SEC.

   compiling local convToFPR : Matrix Expression Integer -> Matrix Fraction Polynomial Integer
Time: 0 SEC.

   compiling local convertToX : List Equation Fraction Polynomial Integer -> List Equation Expression Integer
Time: 0 SEC.

   compiling exported celine : ((Expression Integer,Expression Integer) -> Expression Integer,PositiveInteger,PositiveInteger) -> Union(Matrix Expression Integer,failed)
Time: 0 SEC.

   compiling exported recRel : ((Expression Integer,Expression Integer) -> Expression Integer,PositiveInteger,PositiveInteger) -> Union(Expression Integer,failed)
Time: 0.04 SEC.

   compiling exported recEq : ((Expression Integer,Expression Integer) -> Expression Integer,PositiveInteger,PositiveInteger) -> Union(Equation Expression Integer,failed)
Time: 0.06 SEC.

   compiling exported firstOrderRecurrence : ((Expression Integer,Expression Integer) -> Expression Integer,Expression Integer -> Matrix Expression Integer) -> Expression Integer
Time: 0.20 SEC.

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

   Warnings: 
      [1] sumAndNormalize: 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] sumAndNormalize: 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))

   Cumulative Statistics for Constructor CelineMethod
      Time: 0.39 seconds

   finalizing NRLIB CELINE 
   Processing CelineMethod for Browser database:
--->-->CelineMethod(constructor): Not documented!!!!
--------(coeffMatrix ((Matrix (Expression (Integer))) (PositiveInteger) (PositiveInteger)))---------
--------(funcMatrix ((Matrix (Expression (Integer))) (Mapping (Expression (Integer)) (Expression (Integer)) (Expression (Integer))) (PositiveInteger) (PositiveInteger)))---------
--------(sumAndNormalize ((Expression (Integer)) (Matrix (Expression (Integer))) (Matrix (Expression (Integer)))))---------
--------(retractNumeratorToPolyInt ((Union (Polynomial (Integer)) failed) (Expression (Integer))))---------
--------(findCoeffs ((List (List (Equation (Fraction (Polynomial (Integer)))))) (Polynomial (Integer)) (Matrix (Expression (Integer))) (Symbol)))---------
--------(celine ((Union (Matrix (Expression (Integer))) failed) (Mapping (Expression (Integer)) (Expression (Integer)) (Expression (Integer))) (PositiveInteger) (PositiveInteger)))---------
--------(firstOrderRecurrence ((Expression (Integer)) (Mapping (Expression (Integer)) (Expression (Integer)) (Expression (Integer))) (Mapping (Matrix (Expression (Integer))) (Expression (Integer)))))---------
--------(recRel ((Union (Expression (Integer)) failed) (Mapping (Expression (Integer)) (Expression (Integer)) (Expression (Integer))) (PositiveInteger) (PositiveInteger)))---------
--------(recEq ((Union (Equation (Expression (Integer))) failed) (Mapping (Expression (Integer)) (Expression (Integer)) (Expression (Integer))) (PositiveInteger) (PositiveInteger)))---------
--->-->CelineMethod(): Missing Description
; compiling file "/var/aw/var/LatexWiki/CELINE.NRLIB/CELINE.lsp" (written 31 MAY 2026 03:04:16 PM):

; wrote /var/aw/var/LatexWiki/CELINE.NRLIB/CELINE.fasl
; compilation finished in 0:00:00.068
------------------------------------------------------------------------
   CelineMethod is now explicitly exposed in frame initial 
   CelineMethod will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/CELINE.NRLIB/CELINE

fricas

)version

"FriCAS 1.3.12 compiled at Sat  7 Jun 23:54:49 CEST 2025"

X ==> EXPR INT

Type: Void

fricas

F1:=(n,k)+->binomial(n,k)

$\label{eq1}{\left(n , \: k \right)}\mapsto{binomial \left({n , \: k}\right)}$

(1)

Type: AnonymousFunction?

fricas

r1:=recEq(F1,1,1)

$\label{eq2}{S \left({n}\right)}={2 \ {S \left({n - 1}\right)}}$

(2)

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

fricas

-- Example 4.3.1. 
F2:=(n,k)+->binomial(n,k)^2

$\label{eq3}{\left(n , \: k \right)}\mapsto{{binomial \left({n , \: k}\right)}^{2}}$

(3)

Type: AnonymousFunction?

fricas

r2:=recEq(F2,2,2)

$\label{eq4}{S \left({n}\right)}={\frac{{\left({4 \ n}- 2 \right)}\ {S \left({n - 1}\right)}}{n}}$

(4)

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

fricas

-- Example 4.3.3. 
F3:=(n,k)+->binomial(n,k)*binomial(2*k,k)*(-2)^(n-k)

$\label{eq5}{\left(n , \: k \right)}\mapsto{{binomial \left({n , \: k}\right)}\ {binomial \left({{2 \ k}, \: k}\right)}\ {{\left(- 2 \right)}^{n - k}}}$

(5)

Type: AnonymousFunction?

fricas

r3:=recEq(F3,2,2)

$\label{eq6}{S \left({n}\right)}={\frac{{\left({4 \ n}- 4 \right)}\ {S \left({n - 2}\right)}}{n}}$

(6)

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

fricas

-- Example 4.3.2. 
F4:=(n,k)+->(-1)^k*binomial(n,k)*x^k/factorial(k)

$\label{eq7}{\left(n , \: k \right)}\mapsto{\frac{{{\left(- 1 \right)}^{k}}\ {binomial \left({n , \: k}\right)}\ {{x}^{k}}}{factorial \left({k}\right)}}$

(7)

Type: AnonymousFunction?

fricas

r4:=recEq(F4,2,2)  -- (1,2) already works!

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

(8)

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

fricas

-- Example 4.1.1.
F5:=(n,k)+->k*binomial(n,k)

$\label{eq9}{\left(n , \: k \right)}\mapsto{k \ {binomial \left({n , \: k}\right)}}$

(9)

Type: AnonymousFunction?

fricas

r5:=recEq(F5,1,1)

$\label{eq10}{S \left({n}\right)}={\frac{2 \ n \ {S \left({n - 1}\right)}}{n - 1}}$

(10)

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

fricas

F6:=(n,k)+->k^2*binomial(n,k)

$\label{eq11}{\left(n , \: k \right)}\mapsto{{{k}^{2}}\ {binomial \left({n , \: k}\right)}}$

(11)

Type: AnonymousFunction?

fricas

r6:=recEq(F6,1,1) -- fails, but (1,2) and (2,1) are working .... pref: 1,2

$\label{eq12}\verb#"failed"#$

(12)

Type: Union("failed",...)

fricas

r6_12:=recEq(F6,1,2)

$\label{eq13}{S \left({n}\right)}={\frac{{3 \ n \ {S \left({n - 1}\right)}}-{2 \ n \ {S \left({n - 2}\right)}}}{n - 1}}$

(13)

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

fricas

r6_21:=recEq(F6,2,1)

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

(14)

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

fricas

r6_21 - r612

$\label{eq15}{{S \left({n}\right)}- r 612}={\frac{{{\left({2 \ n}+ 2 \right)}\ {S \left({n - 1}\right)}}+{{\left(- n + 1 \right)}\ r 612}}{n - 1}}$

(15)

Type: Equation(Expression(Integer))