in

Author: PeterLuschny

docs/src/index.md

@@ -1570,6 +1570,9 @@ V035327
 L036038
 ```
 ```@docs
+L036040
+```
+```@docs
 T037027
 ```
 ```@docs
@@ -2206,6 +2209,9 @@ V251568
 V254749
 ```
 ```@docs
+L257490
+```
+```@docs
 V257993
 ```
 ```@docs
@@ -2437,6 +2443,12 @@ I325145
 L325145
 ```
 ```@docs
+L327003
+```
+```@docs
+L327004
+```
+```@docs
 V327491
 ```
 ```@docs

docs/src/modules.md

@@ -216,7 +216,7 @@ P. Luschny, [Generalized Binomial](http://oeis.org/wiki/User:Peter_Luschny/Exten
 [4]   1, 1, 4,  5, 14, 20,  48,  75,  165,  274,  571,  988, ...
 [5]   1, 1, 5,  6, 20, 27,  75, 110,  275,  429, 1001, 1637, ...
 [6]   1, 1, 6,  7, 27, 35, 110, 154,  429,  637, 1638, 2548, ...
-``` 
+```
 
 *  Multinacci, V309896, V006053, V188021, V231181
 
@@ -437,13 +437,13 @@ For example consider the case n = 4. There are five integer partitions of 4:
 
 * P = [[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]]. The shapes are m times the parts of the integer partitions: S(m) = [[4m], [3m, m], [2m, 2m], [2m, m, m], [m, m, m, m]].
 
-* In the case m = 1 we look at set partitions of {1, 2, 3, 4} with sizes in  [[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]] which gives rise to [1, 4, 3, 6, 1] with sum 15.
+* In the case m = 1 we look at set partitions of {1, 2, 3, 4} with sizes ∈  [[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]] which gives rise to [1, 4, 3, 6, 1] with sum 15.
 
-* In the case m = 2 we look at set partitions of {1, 2, .., 8} with sizes in [[8], [6, 2], [4, 4], [4, 2, 2], [2, 2, 2, 2]] which gives rise to [1, 28, 35, 210, 105] with sum 379.
+* In the case m = 2 we look at set partitions of {1, 2, .., 8} with sizes ∈ [[8], [6, 2], [4, 4], [4, 2, 2], [2, 2, 2, 2]] which gives rise to [1, 28, 35, 210, 105] with sum 379.
 
-* In the case m = 0 we look at set partitions of {} with sizes in [[0], [0, 0], [0, 0], [0, 0, 0], [0, 0, 0, 0]] which gives rise to [1, 1, 1, 1, 1] with sum 5 (because the only partition of the empty set is the set that contains the empty set, thus from the definition T(0,4) = Sum_{S(0)} card({0}) = A000041(4) = 5).
+* In the case m = 0 we look at set partitions of {} with sizes ∈ [[0], [0, 0], [0, 0], [0, 0, 0], [0, 0, 0, 0]] which gives rise to [1, 1, 1, 1, 1] with sum 5 (because the only partition of the empty set is the set that contains the empty set, thus from the definition T(0,4) = Sum_{S(0)} card({0}) = A000041(4) = 5).
 
-* If n runs through 0, 1, 2,... then the result is an irregular triangle in which the n-th row lists multinomials for partitions of [m*n] which have only parts which are multiples of m. These are the triangles A080575 (m = 1), A257490 (m = 2), A327003 (m = 3), A327004 (m = 4). In the case m = 0 the triangle is A000012 subdivided into rows of length A000041. See the  references below how this case integrates into the full picture.
+* If n runs through 0, 1, 2,... then the result is an irregular triangle ∈ which the n-th row lists multinomials for partitions of [m*n] which have only parts which are multiples of m. These are the triangles A080575 (m = 1), A257490 (m = 2), A327003 (m = 3), A327004 (m = 4). In the case m = 0 the triangle is A000012 subdivided into rows of length A000041. See the  references below how this case integrates into the full picture.
 
 | type  | m = 0 | m = 1 | m = 2 | m = 3 | m = 4 |
 |-------|-------|-------|-------|-------|-------|
@@ -458,11 +458,7 @@ See also [A260876](https://oeis.org/A260876).
 
    🔶  [SpigotPi](https://github.com/OpenLibMathSeq/IntegerSequences.jl/blob/master/src/SpigotPi.jl)
 
-Computes the first n decimal digits of Pi, uses a variant of the spigot algorithm valid as long as the number of digits <= 54900.
-
-Based on ideas of A. Sale (1968). Algorithm due to D. Saada (1988) and S. Rabinowitz (1991). Proof due to S. Rabinowitz and S. Wagon (1995).
-
-https://www.maa.org/sites/default/files/pdf/pubs/amm_supplements/Monthly_Reference_12.pdf
+Computes the first n decimal digits of Pi, uses a variant of the spigot algorithm valid as long as the number of digits <= 54900. Based on ideas of A. Sale (1968). Algorithm due to D. Saada (1988) and S. Rabinowitz (1991). Proof due to [Rabinowitz and S. Wagon](https://www.maa.org/sites/default/files/pdf/pubs/amm_supplements/Monthly_Reference_12.pdf) (1995).
 
 * Pi, π, L000796
 

src/Abundant.jl

@@ -73,23 +73,23 @@ function demo()
     println(V005101(15))
     println(L005101(15))
 
-    for a in I005101(15)
+    for a ∈ I005101(15)
         print(a, ", ")
     end
     println("...")
-    for a in F005101(40)
+    for a ∈ F005101(40)
         print(a, ", ")
     end
     println("...")
 
-    for n in 40:50
+    for n ∈ 40:50
         println(n, " ↦ ", isAbundant(n))
     end
-    for n in 1:6
+    for n ∈ 1:6
         println(n, " ↦ ", L005101(n))
     end
 
-    for a in I005101(1000)
+    for a ∈ I005101(1000)
         isodd(a) && print(a, " ")
     end
     println()

src/AltPartitions.jl

@@ -54,7 +54,7 @@ function NEXPAR(N::Int, Visit::Function)
     MTC = M[D] ≠ N
     fill!(PAR, 0)
     K = 0
-    for I in 1:D, J in 1:M[I]
+    for I ∈ 1:D, J ∈ 1:M[I]
         K = K + 1
         PAR[K] = R[I]
     end
@@ -146,12 +146,12 @@ function demo()
     # Prints the partitions given in the format used in function NEXPAR.
     PrintPartition(P) =  P |> println
 
-    for n in 1:5
+    for n ∈ 1:5
         VisitPartition(n, PrintPartition)
         println()
     end
 
-    for p in partitions(6)
+    for p ∈ partitions(6)
         Println(p)
     end
     println()

src/AndreNumbers.jl

@@ -35,7 +35,7 @@ function André(m::Int, n::Int)
     haskey(CacheAndré, (m, n)) && return CacheAndré[(m, n)]
     n ≤ 0 && return fmpz(1)
     r = range(0, step = m, stop = n - 1)
-    S = sum(binom(n, k) * André(m, k) for k in r)
+    S = sum(binom(n, k) * André(m, k) for k ∈ r)
     return CacheAndré[(m, n)] = n % m == 0 ? -S : S
 end
 
@@ -91,7 +91,7 @@ C000111() =
             A = 0
             D[k+s] = 0
             s = -s
-            for j in 0:i
+            for j ∈ 0:i
                 A += D[k]
                 D[k] = A
                 k += s
@@ -117,14 +117,14 @@ function test()
 
         if is_oeis_installed()
             V = [V000111, V178963, V178964, V181936, V250283]
-            for v in V
+            for v ∈ V
                 SeqTest(v, 'V')
             end
         end
 
         V = [1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521]
         generator = C000111()
-        for n in 1:10
+        for n ∈ 1:10
             v = take!(generator)
             @test V[n] == v
         end
@@ -137,13 +137,13 @@ function test()
 end
 
 function demo()
-    for m in 1:8
-        Println([André(m, n) for n in 0:11])
+    for m ∈ 1:8
+        Println([André(m, n) for n ∈ 0:11])
     end
 
     println()
     generator = C000111()
-    for n in 0:10
+    for n ∈ 0:10
         v = take!(generator)
         println(n, " ↦ ", v)
     end
@@ -152,12 +152,12 @@ end
 
 """
 
-for m in 1:20, n in 0:100 André(m,n) end
+for m ∈ 1:20, n ∈ 0:100 André(m,n) end
     0.012252 seconds (66.73 k allocations: 1.341 MiB)
 """
 function perf()
     GC.gc()
-    @time (for m in 1:20, n in 0:100
+    @time (for m ∈ 1:20, n ∈ 0:100
         André(m, n)
     end)
 end
@@ -170,7 +170,7 @@ end
 
 main()
 
-for n in 0:32
+for n ∈ 0:32
     println(André(3, n))
 end
 

src/BellNumbers.jl

@@ -39,9 +39,9 @@ function BellNumberList(m::Int)
 
     A = ZArray(m)
     A[1] = fmpz(1)
-    for n in 2:m-1
+    for n ∈ 2:m-1
         A[n] = A[1]
-        for k in n:-1:2
+        for k ∈ n:-1:2
             A[k-1] += A[k]
         end
         R[n+1] = A[1]
@@ -108,7 +108,7 @@ function BellTrans(n::Int, k::Int, X::Array)
         return s
     end
 
-    for m in 1:n-k+1
+    for m ∈ 1:n-k+1
         s += a * BellTrans(n - m, k - 1, X) * X[m]
         a = div(a * (n - m), m)
     end
@@ -137,7 +137,7 @@ function BellTrans(n::Int, k::Int, F::Function)
         return s
     end
 
-    for m in 1:n-k+1
+    for m ∈ 1:n-k+1
         s += a * BellTrans(n - m, k - 1, F) * F(m - 1)
         a = div(a * (n - m), m)
     end
@@ -167,7 +167,7 @@ function BellTriangle(n::Int, seq)
     M = ZTriangle(n)
     i = 1
 
-    for j in 0:n-1, k in 0:j
+    for j ∈ 0:n-1, k ∈ 0:j
         M[i] = BellTrans(j, k, seq)
         i += 1
     end
@@ -400,7 +400,7 @@ function test()
 
     @testset "Bell" begin
         seq = fmpz[1, 1, 2, 5, 15, 52]
-        a = [BellTrans(6, k, seq) for k in 0:5]
+        a = [BellTrans(6, k, seq) for k ∈ 0:5]
         b = [0, 52, 205, 210, 85, 15]
         @test all(a .== b)
 
@@ -410,7 +410,7 @@ function test()
         @test all(a[1:5] .== b[1:5])
 
         a = fmpz[1, 1, 2, 5, 15, 52, 203]
-        b = [BellNumber(n) for n in 0:6]
+        b = [BellNumber(n) for n ∈ 0:6]
         @test all(a .== b)
 
         if is_oeis_installed()
@@ -433,14 +433,14 @@ function demo()
     seq = [1, 1, 2, 5, 15, 52]
     len = size(seq)[1]
 
-    for n in 0:len
-        println([BellTrans(n, k, seq) for k in 0:n])
+    for n ∈ 0:len
+        println([BellTrans(n, k, seq) for k ∈ 0:n])
     end
 
     M = BellTriangle(len, seq)
     ShowAsΔ(M)
 
-    for n in 0:6
+    for n ∈ 0:6
         println(n, " -> ", BellNumberList(n))
     end
 
@@ -451,13 +451,13 @@ end
 """
 
 BellNumberList(1000) :: 0.539088 seconds (855.00 k allocations: 13.062 MiB)
-BellTriangle(100, [1 for _ in 1:100]) :: 0.576952 seconds (1.22 M allocations: 24.571 MB, 14.23% gc time)
+BellTriangle(100, [1 for _ ∈ 1:100]) :: 0.576952 seconds (1.22 M allocations: 24.571 MB, 14.23% gc time)
 """
 function perf()
     GC.gc()
     BellNumberList(5)
     @time BellNumberList(1000)
-    @time BellTriangle(100, [1 for _ in 1:100])
+    @time BellTriangle(100, [1 for _ ∈ 1:100])
 end
 
 function main()