From 193a2ac033b63c26857ca5ce8db9cab3d2934831 Mon Sep 17 00:00:00 2001 From: filifa Date: Wed, 10 Jun 2026 18:42:18 -0400 Subject: [PATCH] add footnote --- junk/faulhaber/faulhaber.html | 23 ++++++++++++++++------- 1 file changed, 16 insertions(+), 7 deletions(-) diff --git a/junk/faulhaber/faulhaber.html b/junk/faulhaber/faulhaber.html index fc5ab78..25d1baf 100644 --- a/junk/faulhaber/faulhaber.html +++ b/junk/faulhaber/faulhaber.html @@ -20,6 +20,7 @@

Sums of Powers

Posted

+

Updated

This page is all about how we can efficiently compute large sums of @@ -402,7 +403,7 @@

This algorithm is quadratic in the value of p, which isn't great, but it's simple to - implement (and I don't know of any faster way).

+ implement.1

Stirling numbers calculator

@@ -460,7 +461,7 @@

It turns out there is a very elegant identity relating exponents, Stirling numbers, and falling - factorials:1

+ factorials:2

@@ -836,8 +837,8 @@

(For p=0, the sum is - just n.)2 This lets us - simplify.3

+ just n.)3 This lets us + simplify.4

@@ -1057,13 +1058,21 @@

Footnotes

    -
  1. 1 For proofs of this identity, see chapter 1.9 of +
  2. 1 I think we can technically compute all the Stirling + numbers we need in linearithmic time by applying the convolution + theorem to the numbers' explicit sum formula, but that approach + makes it difficult to get exact values. If you just want the + Stirling numbers modulo a prime, you could maybe use a + number-theoretic transform, + but that's well outside my wheelhouse so I'm not going to touch + on that here.
    3. +
  3. 2 For proofs of this identity, see chapter 1.9 of Enumerative Combinatorics by Stanley and chapter 6.1 of Concrete Mathematics by Graham, Knuth, and Patashnik.
    4. -
  4. 2 See chapter 2.6 of Concrete +
  5. 3 See chapter 2.6 of Concrete Mathematics for an explanation of this identity.
    6. -
  6. 3 A version of this formula is also mentioned in +
  7. 4 A version of this formula is also mentioned in chapter 6.5 of Concrete Mathematics.