gaus

Author: PeterLuschny

docs/src/modules.md

@@ -25,7 +25,7 @@ Generalized André numbers count the ``m``-alternating permutations of length ``
 
 The Bell transform transforms an integer sequence into an integer triangle; also known as incomplete Bell polynomials.
 
-Let ``X`` be an integer sequence, then ``B_{n, k}(X) = \\sum_{m=1}^{n-k+1} \\binom{n-1}{m-1} X[m] B_{n-m,k-1}(X)`` where ``B_{0,0} = 1, B_{n,0} = 0`` for ``n≥1, B_{0,k} = 0`` for ``k≥1``.
+Let ``X`` be an integer sequence, then ``B_{n, k}(X) = \sum_{m=1}^{n-k+1} \binom{n-1}{m-1} X[m] B_{n-m,k-1}(X)`` where ``B_{0,0} = 1, B_{n,0} = 0`` for ``n≥1, B_{0,k} = 0`` for ``k≥1``.
 
 The Bell transform is (0,0)-based and the associated triangle always has as first column 1,0,0,0,... This column is often missing in the OEIS. Other Stirling number related sequences are implemented in the module StirlingLahNumbers.
 
@@ -54,7 +54,7 @@ The rational Bernoulli numbers are defined here with ``B(1) = 1/2``. Why this is
 
    🔶  [BinaryInteger](https://github.com/OpenLibMathSeq/IntegerSequences.jl/blob/master/src/BinaryInteger.jl)
 
-For positive n, BinaryIntegerLength is ``\\lfloor(\\operatorname{log}(2, n)\\rfloor) + 1``, BinaryIntegerLength(0) = 0.
+For positive n, BinaryIntegerLength is ``⌊ log[2](n) ⌋ + 1``, BinaryIntegerLength(0) = 0.
 
 * BinaryIntegerLength, Bil, V001855, V003314, V033156, V054248, V061168, V083652, V097383, V123753, V295513
 
@@ -70,10 +70,10 @@ Ported from [BinaryQuadraticForms](http://oeis.org/wiki/User:Peter_Luschny/Binar
 
    🔶  [CantorMachines](https://github.com/OpenLibMathSeq/IntegerSequences.jl/blob/master/src/CantorMachines.jl)
 
-* Cantor-Machine, Cantor-Enumeration, Cantor-Pairing, Cantor-BoustrophedonicMachine, Cantor-BoustrophedonicEnumeration, Cantor-BoustrophedonicPairing, RosenbergStrong-BoustrophedonicMachine, RosenbergStrong-BoustrophedonicEnumeration, RosenbergStrong-BoustrophedonicPairing
-
 [Cantor's enumeration of N X N revisited](https://luschny.wordpress.com/2018/09/24/cantors-enumeration-of-n2-revisited/).
 
+* Cantor-Machine, Cantor-Enumeration, Cantor-Pairing, Cantor-BoustrophedonicMachine, Cantor-BoustrophedonicEnumeration, Cantor-BoustrophedonicPairing, RosenbergStrong-BoustrophedonicMachine, RosenbergStrong-BoustrophedonicEnumeration, RosenbergStrong-BoustrophedonicPairing
+
    🔶  [CarmichaelNumbers](https://github.com/OpenLibMathSeq/IntegerSequences.jl/blob/master/src/CarmichaelNumbers.jl)
 
 * isCarmichael, I002997, F002997, L002997

src/BernoulliNumbers.jl

@@ -32,7 +32,7 @@ The rational Bernoulli numbers are defined here with ``B(1) = 1/2``. Why this is
 const ModuleBernoulliNumbers = ""
 
 """
-Return the generalized integer Bernoulli numbers ``b_{m}(n) = n \\times ``André``(m, n-1)``.
+Return the generalized integer Bernoulli numbers ``b_{m}(n) = n × ``André``(m, n-1)``.
 """
 BernoulliInt(m::Int, n::Int) = n == 0 ? ZZ(0) : n * André(m, n - 1)
 
@@ -42,12 +42,12 @@ Return the number of down-up permutations w on ``[n+1]`` such that ``w_2 = 1``.
 V065619(n::Int) = BernoulliInt(2, n)
 
 """
-Return the generalized integer Bernoulli numbers ``b_{3}(n) = n \\times ``André``_{3}(n-1)``.
+Return the generalized integer Bernoulli numbers ``b_{3}(n) = n × ``André``_{3}(n-1)``.
 """
 V281586(n::Int) = BernoulliInt(3, n)
 
 """
-Return the generalized integer Bernoulli numbers ``b_{4}(n) = n \\times `` André``_{4}(n-1)``.
+Return the generalized integer Bernoulli numbers ``b_{4}(n) = n × `` André``_{4}(n-1)``.
 """
 V281588(n::Int) = BernoulliInt(4, n)
 

src/BinaryInteger.jl

@@ -10,14 +10,14 @@ export V001855, V003314, V033156, V054248, V061168, V083652, V097383, V123753
 export V295513, BinaryIntegerLength, Bil
 
 """
-For positive n, BinaryIntegerLength is ``\\lfloor(\\operatorname{log}(2, n)\\rfloor) + 1``, BinaryIntegerLength(0) = 0.
+For positive n, BinaryIntegerLength is ``⌊ log[2](n) ⌋ + 1``, BinaryIntegerLength(0) = 0.
 
 * BinaryIntegerLength, Bil, V001855, V003314, V033156, V054248, V061168, V083652, V097383, V123753, V295513
 """
 const ModuleBinaryInteger = ""
 
 """
-Return the length of the binary extension of an integer ``n``, which is defined as ``0`` if ``n = 0`` and for ``n > 0`` as ``⌊ log _2(n) ⌋ + 1``.
+Return the length of the binary extension of an integer ``n``, which is defined as ``0`` if ``n = 0`` and for ``n > 0`` as ``⌊ log[2](n) ⌋ + 1``.
 """
 BinaryIntegerLength(n) = n == 0 ? 0 : floor(Int, log2(n)) + 1
 
@@ -62,7 +62,7 @@ Minimum total number of comparisons to find each of the values ``1`` through ``n
 V097383(n) = V295513(n+1) - div(n-1, 2)
 
 """
-Partial sums of the sequence ``⌊ \\log_2(n) ⌋``.
+Partial sums of the sequence ``⌊ log[2](n) ⌋``.
 """
 V061168(n) = V295513(n+1) - n + 1
 

src/BuildSequences.jl

@@ -394,6 +394,7 @@ function make_modules()
                 name = splitext(filename)
                 println(mod, "\n   🔶  ", '[', name[1], "](", path, ")\n" )
             else
+                n = replace(n, "\\\\" => "\\")
                 indoc && println(mod, n)
             end
         end

src/CantorMachines.jl

@@ -12,9 +12,9 @@ export RosenbergStrongBoustrophedonicMachine, RosenbergStrongBoustrophedonicEnum
 export V319514, L319514
 
 """
-* Cantor-Machine, Cantor-Enumeration, Cantor-Pairing, Cantor-BoustrophedonicMachine, Cantor-BoustrophedonicEnumeration, Cantor-BoustrophedonicPairing, RosenbergStrong-BoustrophedonicMachine, RosenbergStrong-BoustrophedonicEnumeration, RosenbergStrong-BoustrophedonicPairing
-
 [Cantor's enumeration of N X N revisited](https://luschny.wordpress.com/2018/09/24/cantors-enumeration-of-n2-revisited/).
+
+* Cantor-Machine, Cantor-Enumeration, Cantor-Pairing, Cantor-BoustrophedonicMachine, Cantor-BoustrophedonicEnumeration, Cantor-BoustrophedonicPairing, RosenbergStrong-BoustrophedonicMachine, RosenbergStrong-BoustrophedonicEnumeration, RosenbergStrong-BoustrophedonicPairing
 """
 const ModuleCantorMachines = ""
 

src/IntegerSequences.jl

@@ -2,8 +2,8 @@
 # Copyright Peter Luschny. License is MIT.
 # This file includes parts from Combinatorics.jl in modified form.
 
-# Version of: UTC 2019-05-26 09:11:30
-# 7c6236a0-7f85-11e9-0f1c-7dc10fb521d5
+# Version of: UTC 2019-05-26 10:30:37
+# 89d7f300-7f90-11e9-2316-5b680c2c6b88
 
 # Do not edit this file, it is generated from the modules and will be overwritten!
 # Edit the modules in the src directory and build this file with BuildSequences.jl!
@@ -1025,7 +1025,7 @@ The rational Bernoulli numbers are defined here with ``B(1) = 1/2``. Why this is
 """
 const ModuleBernoulliNumbers = ""
 """
-Return the generalized integer Bernoulli numbers ``b_{m}(n) = n \\times ``André``(m, n-1)``.
+Return the generalized integer Bernoulli numbers ``b_{m}(n) = n × ``André``(m, n-1)``.
 $(SIGNATURES)
 """
 BernoulliInt(m::Int, n::Int) = n == 0 ? ZZ(0) : n * André(m, n - 1)
@@ -1035,12 +1035,12 @@ $(SIGNATURES)
 """
 V065619(n::Int) = BernoulliInt(2, n)
 """
-Return the generalized integer Bernoulli numbers ``b_{3}(n) = n \\times ``André``_{3}(n-1)``.
+Return the generalized integer Bernoulli numbers ``b_{3}(n) = n × ``André``_{3}(n-1)``.
 $(SIGNATURES)
 """
 V281586(n::Int) = BernoulliInt(3, n)
 """
-Return the generalized integer Bernoulli numbers ``b_{4}(n) = n \\times `` André``_{4}(n-1)``.
+Return the generalized integer Bernoulli numbers ``b_{4}(n) = n × `` André``_{4}(n-1)``.
 $(SIGNATURES)
 """
 V281588(n::Int) = BernoulliInt(4, n)
@@ -1147,13 +1147,13 @@ P = Primes(2, div(n + 2, 2 + n % 2))
 end
 # *** BinaryInteger.jl ****************
 """
-For positive n, BinaryIntegerLength is ``\\lfloor(\\operatorname{log}(2, n)\\rfloor) + 1``, BinaryIntegerLength(0) = 0.
+For positive n, BinaryIntegerLength is ``⌊ log[2](n) ⌋ + 1``, BinaryIntegerLength(0) = 0.
 
 * BinaryIntegerLength, Bil, V001855, V003314, V033156, V054248, V061168, V083652, V097383, V123753, V295513
 """
 const ModuleBinaryInteger = ""
 """
-Return the length of the binary extension of an integer ``n``, which is defined as ``0`` if ``n = 0`` and for ``n > 0`` as ``⌊ log _2(n) ⌋ + 1``.
+Return the length of the binary extension of an integer ``n``, which is defined as ``0`` if ``n = 0`` and for ``n > 0`` as ``⌊ log[2](n) ⌋ + 1``.
 $(SIGNATURES)
 """
 BinaryIntegerLength(n) = n == 0 ? 0 : floor(Int, log2(n)) + 1
@@ -1198,7 +1198,7 @@ $(SIGNATURES)
 """
 V097383(n) = V295513(n+1) - div(n-1, 2)
 """
-Partial sums of the sequence ``⌊ \\log_2(n) ⌋``.
+Partial sums of the sequence ``⌊ log[2](n) ⌋``.
 $(SIGNATURES)
 """
 V061168(n) = V295513(n+1) - n + 1
@@ -1599,9 +1599,9 @@ $(SIGNATURES)
 L007522(bound::Int) = binaryQF(-1, 4, 4, bound, "prime")
 # *** CantorMachines.jl ****************
 """
-* Cantor-Machine, Cantor-Enumeration, Cantor-Pairing, Cantor-BoustrophedonicMachine, Cantor-BoustrophedonicEnumeration, Cantor-BoustrophedonicPairing, RosenbergStrong-BoustrophedonicMachine, RosenbergStrong-BoustrophedonicEnumeration, RosenbergStrong-BoustrophedonicPairing
-
 [Cantor's enumeration of N X N revisited](https://luschny.wordpress.com/2018/09/24/cantors-enumeration-of-n2-revisited/).
+
+* Cantor-Machine, Cantor-Enumeration, Cantor-Pairing, Cantor-BoustrophedonicMachine, Cantor-BoustrophedonicEnumeration, Cantor-BoustrophedonicPairing, RosenbergStrong-BoustrophedonicMachine, RosenbergStrong-BoustrophedonicEnumeration, RosenbergStrong-BoustrophedonicPairing
 """
 const ModuleCantorMachines = ""
 """
@@ -4536,12 +4536,12 @@ $(SIGNATURES)
 """
 T216916(dim::Int) = OrthoPoly(dim, n -> n + 1, n -> n + 1)
 """
-Return the triangle ``T(n,k)`` of tangent numbers, coefficient of ``x^n/n!`` in the expansion of ``(\\tan x)^k/k!``.
+Return the triangle ``T(n,k)`` of tangent numbers, coefficient of ``x^n/n!`` in the expansion of ``(tan x)^k/k!``.
 $(SIGNATURES)
 """
 T059419(dim::Int) = OrthoPoly(dim, n -> 0, n -> n * (n-1))
 """
-Return the expansion of ``\\exp(\\tan(x))``.
+Return the expansion of exp(tan(x)).
 $(SIGNATURES)
 """
 L006229(len::Int) = RowSums(T059419(len))

src/OrthoPolynomials.jl

@@ -74,12 +74,12 @@ Return the coefficients of some orthogonal polynomials related to set partitions
 T216916(dim::Int) = OrthoPoly(dim, n -> n + 1, n -> n + 1)
 
 """
-Return the triangle ``T(n,k)`` of tangent numbers, coefficient of ``x^n/n!`` in the expansion of ``(\\tan x)^k/k!``.
+Return the triangle ``T(n,k)`` of tangent numbers, coefficient of ``x^n/n!`` in the expansion of ``(tan x)^k/k!``.
 """
 T059419(dim::Int) = OrthoPoly(dim, n -> 0, n -> n * (n-1))
 
 """
-Return the expansion of ``\\exp(\\tan(x))``.
+Return the expansion of exp(tan(x)).
 """
 L006229(len::Int) = RowSums(T059419(len))
 

test/perftests.jl

@@ -2,8 +2,8 @@
 # Copyright Peter Luschny. License is MIT.
 # This file includes parts from Combinatorics.jl in modified form.
 
-# Version of: UTC 2019-05-26 09:11:30
-# 7c693b80-7f85-11e9-0d44-11817a2a065c
+# Version of: UTC 2019-05-26 10:30:37
+# 89df4600-7f90-11e9-3bf6-c39fb02ccd92
 
 # Do not edit this file, it is generated from the modules and will be overwritten!
 # Edit the modules in the src directory and build this file with BuildSequences.jl!

test/runtests.jl

@@ -2,8 +2,8 @@
 # Copyright Peter Luschny. License is MIT.
 # This file includes parts from Combinatorics.jl in modified form.
 
-# Version of: UTC 2019-05-26 09:11:30
-# 7c6766c0-7f85-11e9-28eb-3741150190c5
+# Version of: UTC 2019-05-26 10:30:37
+# 89dc11ae-7f90-11e9-1984-5170e93e3915
 
 # Do not edit this file, it is generated from the modules and will be overwritten!
 # Edit the modules in the src directory and build this file with BuildSequences.jl!