From 97620f134cf945bfbe7dc2e5c54f467fec3f3a0c Mon Sep 17 00:00:00 2001
From: thundergnat <thundergnat@comcast.net>
Date: Thu, 29 Oct 2020 19:35:42 -0400
Subject: [PATCH 1/2] Fix spelling of Sterling -> Stirling

---
 README.md                      |  4 ++--
 lib/Math/Sequences/Integer.pm6 | 24 ++++++++++++------------
 2 files changed, 14 insertions(+), 14 deletions(-)

diff --git a/README.md b/README.md
index 6546380..1ebbdd1 100644
--- a/README.md
+++ b/README.md
@@ -84,9 +84,9 @@ To gain access to these, use:
 * `sigma($n, $exponent=1)`
   The sum of positive divisors function σ. The optional exponent is
   the power to which each divisor is raised before summing.
-* `Sterling1($n, $k)`
+* `Stirling1($n, $k)`
   Count permutations according to their number of cycles.
-* `Sterling2($n, $k)`
+* `Stirling2($n, $k)`
   The number of ways to partition a set of n objects into k non-empty subsets.
 * `totient($n)`
   The numbers from zero to n that are co-prime to n.
diff --git a/lib/Math/Sequences/Integer.pm6 b/lib/Math/Sequences/Integer.pm6
index 3e0052d..718c88d 100644
--- a/lib/Math/Sequences/Integer.pm6
+++ b/lib/Math/Sequences/Integer.pm6
@@ -337,20 +337,20 @@ sub Eulers-number ( Int $terms = 500 ) is export(:support) {
 }
 
 # Stirling numbers of the first kind
-multi Sterling1 (0, 0) is export(:support) { 1 }
-multi Sterling1 (Int \n where * > 0, 0) is export(:support) { 0 }
-multi Sterling1 ( 0, Int \k where * > 0) is export(:support) { 0 }
-multi Sterling1 (Int \n, Int \k) is export(:support) {
+multi Stirling1 (0, 0) is export(:support) { 1 }
+multi Stirling1 (Int \n where * > 0, 0) is export(:support) { 0 }
+multi Stirling1 ( 0, Int \k where * > 0) is export(:support) { 0 }
+multi Stirling1 (Int \n, Int \k) is export(:support) {
     state %seen;
-    (%seen{"{n - 1}|{k - 1}"} //= Sterling1(n - 1, k - 1)) -
-    (n - 1) * (%seen{"{n - 1}|{k}"} //= Sterling1(n - 1, k))
+    (%seen{"{n - 1}|{k - 1}"} //= Stirling1(n - 1, k - 1)) -
+    (n - 1) * (%seen{"{n - 1}|{k}"} //= Stirling1(n - 1, k))
 }
 
 # Stirling numbers of the second kind
-multi Sterling2 (0, 0) is export(:support) { 1 }
-multi Sterling2 (Int \n where * > 0, 0) is export(:support) { 0 }
-multi Sterling2 (Int \n, Int \k where * == n) is export(:support) { 1 }
-multi Sterling2 (Int \n, Int \k) is export(:support) {
+multi Stirling2 (0, 0) is export(:support) { 1 }
+multi Stirling2 (Int \n where * > 0, 0) is export(:support) { 0 }
+multi Stirling2 (Int \n, Int \k where * == n) is export(:support) { 1 }
+multi Stirling2 (Int \n, Int \k) is export(:support) {
     1/factorial(k) * sum (0 .. k).map: -> \j {
         (-1)**j * (k choose j) * (k - j)**n
     }
@@ -863,9 +863,9 @@ our @A006966 is export = 1, 1, 1, 1, 2, 5, 15, 53, 222, 1078, 5994, 37622,
 # A007318 / Pascal's triangle
 our @A007318 is export = 𝕀.triangle.map: -> ($n,$k) { $n choose $k };
 # A008275 / Stirling 1
-our @A008275 is export = flat (1..*).map: -> $s { (1..$s).map: { Sterling1($s, $_) } };
+our @A008275 is export = flat (1..*).map: -> $s { (1..$s).map: { Stirling1($s, $_) } };
 # A008277 / Stirling 2
-our @A008277 is export = flat (1..*).map: -> $s { (1..$s).map: { Sterling2($s, $_) } };
+our @A008277 is export = flat (1..*).map: -> $s { (1..$s).map: { Stirling2($s, $_) } };
 # A008279 / permutations k at a time
 our @A008279 is export = 𝕀.triangle.map: -> ($n,$k) {
     factorial($n)/factorial($n-$k);

From ee95620ca3c390399740e9625bbd0906d81fcd2c Mon Sep 17 00:00:00 2001
From: thundergnat <thundergnat@comcast.net>
Date: Thu, 29 Oct 2020 19:48:36 -0400
Subject: [PATCH 2/2] Add a generator function for @A000041 / partitions

---
 lib/Math/Sequences/Integer.pm6 | 8 +++++++-
 1 file changed, 7 insertions(+), 1 deletion(-)

diff --git a/lib/Math/Sequences/Integer.pm6 b/lib/Math/Sequences/Integer.pm6
index 718c88d..48bbf8e 100644
--- a/lib/Math/Sequences/Integer.pm6
+++ b/lib/Math/Sequences/Integer.pm6
@@ -417,7 +417,13 @@ our @A000035 is export = |(0,1) xx *;
 # A000040 / primes
 our @A000040 is export = lazy 𝕀.grep: {.is-prime};
 # A000041 / partitions
-our @A000041 is export = 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, &NOSEQ ... *;
+our @A000041 is export = 1, { partition-sum(++$) } … *;
+my @A000041-adder = lazy [\+] flat 1, ( (1 .. *) Z (1 .. *).map: * × 2 + 1 );
+sub partition-sum ($n) {
+    sum @A000041[$n X- @A000041-adder[^(@A000041-adder.first: * > $n, :k)]]
+        Z× (flat (1, 1, -1, -1) xx *)
+}
+
 # A000043 / Mersenne
 our @A000043 is export = lazy 𝕀.grep: { .is-prime and (2**$_-1).is-prime };
 # A000045 / Fibonacci