Library

Supertype for sequences (or sequence-like types).

Supertype for triangles (or triangles-like types).

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The AltSum of a ℤSeq is the alternating sum.

julia> AltSum([0, 1, 2, 3, 4, 5])
+ 0 - 1 + 2 - 3 + 4 - 5 = 6 - 9 = - 3

The AltSum of a ℤTri is the sequence of the alternating sums of the rows.

AltSum(T::ℤTri) = EvenSum(T) - OddSum(T)

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The Central of a ℤTri is the sequence of the middle term of the even indexed rows, indexing starts with 0.

Central(T::ℤTri) = Middle.(T[begin:2:end])

Return the list of coefficients of the polynomial p (ordered by ascending powers).

julia> R, x = ZPolyRing("x")
p = 1 + 2*x + 3*x^2
Coefficients(p)

Return the sequence of list of coefficients of the polynomials P.

Return the convolution triangle of T.

julia> T = [ℤInt[1], ℤInt[2, 3], ℤInt[4, 5, 6], ℤInt[7, 8, 9, 10]]
Println.(ConvolutionTriangle(T))
[1]
[4, 12]
[16, 40, 73]
[49, 112, 190, 284]

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The DiagSum of a ℤTri is the sum of the diagonal triangle.

DiagSum(T::ℤTri) = sum(DiagonalTriangle(T))

Return the diagonal triangle T(n - k, k) where k in 0:n/2.

julia> Println.(DiagonalTriangle(MotzkinTriangle(8)))
[1]
[1]
[2, 1]        
[4, 2]        
[9, 5, 1]     
[21, 12, 3]   
[51, 30, 9, 1]
[127, 76, 25, 4]  # A106489

Return the Laplace expansion of a bivariate exponential generating function as a power series.

Return the Laplace expansion of a bivariate exponential generating function as a list of the coefficients of the power series.

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Return the value of the ℤ-polynomial p evaluated at x.

julia> R, x = ZPolyRing("x")
p = 1 + 2*x + 3*x^2
Evaluate(p, 2)
17

Return the sequence of values to which the sequence of ℤ-polynomials P evaluate at x.

julia> R, x = ZPolyRing("x")
P = [sum(k * x^k for k in 0:n) for n in 0:9]
Evaluate(P, 3)  # A289399

The EvenSum of a ℤSeq is the sum of the even indexed terms, indexing starts with 0.

julia> EvenSum([0, 1, 2, 3, 4, 5])
0 + 2 + 4 = 6

The EvenSum of a ℤTri is the sequence of the even sums of the rows.

EvenSum(T::ℤTri) = EvenSum.(T)

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Return the factorial $n! = ∏(1, n)$.

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Alias for InverseTriangle

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The LeftSide of a ℤSeq is the first term.

julia> LeftSide([0, 1, 2, 3, 4, 5])
0

The LeftSide of a ℤTri is the sequence of the first term in the rows.

LeftSide(T::ℤTri) = LeftSide.(T)

LinMap(M::ℤTri, V::ℤSeq) returns the 'matrix times vector' product of M and V. Expands up to length(V) and we assume length(M) >= length(V).

LinMap(F::Function, V::ℤSeq) returns the 'matrix times vector' product of a matrix generated by F and V.

julia>  L(n) = [ZZ(k) for k in 0:n]
[LinMap(L, L(n)) for n in 0:9]
0
1
5
14
30
... # A000330

The Middle of a ℤSeq A is the middle term, A[div(begin + end, 2)].

julia> Middle([0, 1, 2, 3, 4, 5])
3

The Middle of a ℤTri is the sequence of the middle term in the rows.

Middle(T::ℤTri) = Middle.(T)

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Return Numerator((-2)^degree(p) * Evaluate(p, QQ(-1, 2)))

julia> R, x = ZPolyRing("x")
p = 1 + 2*x + 3*x^2
NegHalf(p)
3

Return the sequence generated by broadcasting NegHalf over the rows of the triangle interpreted as a polynomials.

NegHalf(T::ℤTri) = NegHalf.(Polynomial(T))

Return the sequence generated by broadcasting NegHalf over the sequence of polynomials.

The OddSum of a ℤSeq is the sum of the odd indexed terms, indexing starts with 0.

julia> OddSum([0, 1, 2, 3, 4, 5])
1 + 3 + 5 = 9

The OddSum of a ℤTri is the sequence of the odd sums of the rows.

OddSum(T::ℤTri) = OddSum.(T)

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Return the ℤ-polynomial whose coefficients are the terms of the sequence.

[1, 2, 3] -> 1 + 2*x + 3*x^2

Return the sequence of ℤ-polynomials whose coefficients are the terms of the triangle.

Polynomial(T::ℤTri) = Polynomial.(T)

Return the polynomial function associated with the polynomial with coefficients given by the sequence S. A polynomial function evaluates to Float64 values.

p = PolynomialFunction([1, 2, 3])
julia> [p(r) for r in 0:3]
4-element Vector{Float64}:
  1.0
  6.0
 17.0
 34.0

Return the sequence of polynomial functions associated to the polynomials with coefficients given by the triangle T.

PolynomialFunction(T::ℤTri) = PolynomialFunction.(T)

Return numerator(2^degree(p) * Evaluate(p, QQ(1, 2))).

julia> R, x = ZPolyRing("x")
p = 1 + 2*x + 3*x^2
PosHalf(p)
11

Return the sequence generated by broadcasting PosHalf over the rows of the triangle interpreted as a polynomials.

julia> T = [[ZZ(k) for k in 0:n] for n in 1:5]
PosHalf(Polynomial(T))  # A000295
5-element ℤSeq
 1
 4
 11
 26
 57

Return the sequence generated by broadcasting PosHalf over the sequence of polynomials.

Print the array without typeinfo.

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Constructor for an ℚSeq of length len. If a second parameter f is given the sequence is constructed as [QQ(f(n)) for n in 0:len-1]

Constructor for QTri.

A recursive triangle RecTriangle is a subtype of AbstractTriangle. The rows of the triangle are generated by a function gen(n, k, prevrow) defined for $n ≥ 0$ and $0 ≤ k ≤ n$. The function returns value of type ℤInt.

The parameter prevrow is a function which returns the values of row(n-1) of the triangle and 0 if $k < 0$ or $k > n$. The function prevrow is provided by an instance of RecTriangle and must not be defined by the user.

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Return the row reversed triangle.

Return the ℤ-polynomial whose coefficients are the terms of the reversed sequence.

[1, 2, 3] -> x^2 + 2*x + 3

Return the sequence of ℤ-polynomials whose coefficients are the terms of the reversed triangle.

ReversePolynomial(T::ℤTri) = ReversePolynomial.(T)

The RightSide of a ℤSeq is the last term.

julia> RightSide([0, 1, 2, 3, 4, 5])
5

The RightSide of a ℤTri is the sequence of the last term in the rows.

RightSide(T::ℤTri) = RightSide.(T)

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Print an integer triangle without typeinfo.

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Return an iterator expanding the given sequence to a regular triangle.

julia> T = Telescope(ℤInt[1, 2, 3, 4, 5, 6])
collect(T)
 [1]
 [1, 2]
 [1, 2, 3]
 [1, 2, 3, 4]
 [1, 2, 3, 4, 5]
 [1, 2, 3, 4, 5, 6]

Return an iterator expanding the sequence generated by the function f to a regular triangle.

Prints a list of traits of the triangle with Anums.

Prints a list of traits of the triangle without Anums and, if assoc=true, also a list of associated triangles.

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Transform a ℤSeq V by the matrix/vector product by applying LinMap to (M, V). Expands up to min(length(M), length(V)).

TransAlts(T) = Trans(T, [(-1)^n * ZZ(1) for n = 0:length(T)])

Standard trait of T.

TransNat0(T) = Trans(T, [ZZ(n) for n = 0:length(T)])

Standard trait of T.

TransNat1(T) = Trans(T, [ZZ(n) for n = 1:length(T)])

Standard trait of T.

TransSqrs(T) = Trans(T, [ZZ(n^2) for n = 0:length(T)])

Standard trait of T.

Print the standard traits generated by linear transformation.

julia> TransTraits(BinomialTriangle)
BinomialTriangle
TransSqrs: A001788 [0, 1, 6, 24, 80, 240, 672, 1792, 4608, 11520]
TransNat0: A001787 [0, 1, 4, 12, 32, 80, 192, 448, 1024, 2304]
TransNat1: A001792 [1, 3, 8, 20, 48, 112, 256, 576, 1280, 2816]

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Constructor for an ℤSeq of length len. If a second parameter f is given the sequence is constructed as [ZZ(f(n)) for n in 0:len-1]

Constructor for ZTri.

Search the OEIS for a sequence. The file is saved in the 'data' directory in json format.

ℚInt is an alias for the type Nemo.fmpq.

ℚPolySeq is an alias for the type Array{Nemo.fmpq_poly, 1}.

ℚSeq is an alias for the type Array{Nemo.fmpq, 1}.

ℚTri is an alias for the type Array{Array{Nemo.fmpq, 1}, 1}.

ℤInt is an alias for the type Nemo.fmpz.

ℤPolySeq is an alias for the type Array{Nemo.fmpz_poly, 1}.

ℤSeq is an alias for the type Array{Nemo.fmpz, 1}.

ℤTri is an alias for the type Array{Array{Nemo.fmpz, 1}, 1}.

Return the up-down numbers (2-alternating permutations).

Iterates over the first n rows of A008279.

Iterates over the first n rows of A048993. Triangle of Stirling numbers of 2nd kind.

Iterates over the first n rows of A094587.

xIterates over the first n rows of A132393. Triangle of unsigned Stirling numbers of the first kind.