Phi operator taken from: An informal introduction to the negative rows of Pascal's triangle, discussing the motivation and intuition behind some of its basic applications, before diving into discrete calculus, the forward and backward Newton series, umbral calculus, and as a consequence, a quick overview of Stirling numbers, for the purpose of highlighting some deeper, underappreciated treatments of discrete math. Technical errors: 15:16, the last coefficient in the formula should be 1, not 3. 37:02, the falling and rising factorial are products, and should not have plus signs next to the ellipsis. 39:07, the rising factorial was not substituted correctly, the (-1) factors should not be present in the equation at this step. Supplement material: Finite Calculus: A Tutorial for Solving Nasty Sums by David Gleich: %202005%20-%20finite% Calculus Of Finite Differences (Fourth Edition) by George Boole: What is the sequence at 9:51? Foundational concepts: Similar videos: Further study: Conclusion poster: Petition for a math video you would like to see: My Patreon: 0:00 Overview/Introduction 0:40 Quick review of Pascal's triangle 2:22 Chapter 1: The dark side of Pascal's triangle 9:45 Chapter 2: Finite differences 16:31 Chapter 3: Combinatorial identities 20:03 Chapter 4: Discrete calculus 31:43 Chapter 5: The dark portal 36:23 Chapter 6: Umbral calculus 47:00 What did we learn? / Conclusion 50:55 Final comments and outro Intro riff taken from: Nikolai Kapustin - 8 Concert Etudes, Op. 40: III. Toccatina Music Credit: Sunday / Beyza Quiet Devotion / Martin Klem Ever After / William Claeson Underneath the Christmas Stars / Martin Landh Meadow Breeze / William Claeson Meant to Love / William Benckert Sen / Trevor Kowalski A Slow Drift / Hanna Lindgren Memory of Perception / Rand Aldo Lone Lights / Blue Saga The Choices We Make / Hanna Lindgren Late Nights in Harmony / Trevor Kowalski Turtle Swim / Johannes Bornlof and yos / By Lotus Dismantle / Peter Sandberg For Want of Means / Jay Winters When Sun Meets Moon / Gavin Luke vallmo / baegel courtesy of