Stirling Numbers and Pascal Triangles - Foundations of Linear Algebra

Stirling Numbers and Pascal Triangles - Foundations of Linear Algebra

Insights into Mathematics via YouTube Direct link

CONTENT SUMMARY: pg 1: @00:08 Intro: Stirling numbers and Pascal triangles; sequences; change of terminology @00:44 ; falling power; rising power; list of rising powers; summation notation and Stirliā€¦

1 of 28

1 of 28

CONTENT SUMMARY: pg 1: @00:08 Intro: Stirling numbers and Pascal triangles; sequences; change of terminology @00:44 ; falling power; rising power; list of rising powers; summation notation and Stirliā€¦

Class Central Classrooms beta

YouTube videos curated by Class Central.

Classroom Contents

Stirling Numbers and Pascal Triangles - Foundations of Linear Algebra

Automatically move to the next video in the Classroom when playback concludes

  1. 1 CONTENT SUMMARY: pg 1: @00:08 Intro: Stirling numbers and Pascal triangles; sequences; change of terminology @00:44 ; falling power; rising power; list of rising powers; summation notation and Stirliā€¦
  2. 2 pg 2: @04:55 James Stirling 1749, "Methodus Differentialis"; Stirling number notation warning @05:04 ; 'n bracket k' as Karamata notation Knuth; Stirling numbers of the first kind; Change of basis reā€¦
  3. 3 pg 3: @ Calculating Stirling numbers; Theorem Recurrence relation: Stirling numbers; proof;
  4. 4 pg 4: @ Pascal's triangle and binomial coefficients; recurrence relation for binomial coefficients; Pascal matrix;
  5. 5 pg 5: @ Combinatorial interpretation of Sterling numbers;
  6. 6 pg 6: @17:34 Number theoretic interpretation of Sterling numbers; summary of Sterling number interpretation @;
  7. 7 pg 7: @ Sterling numbers of the 2nd kind; Inverting the Pascal matrices;
  8. 8 pg 8: @26:36 Inverting Stirling matrices; reintroduction of some ignored symmetry @ ; Sterling matrix of the 2nd kind;
  9. 9 pg 9: @ Definition of Stirling numbers of the second kind; 'n brace k' notation of Stirling numbers of the 2nd kind; Sterling matrix of the 2nd kind;
  10. 10 pg 10: @ Combinatorial interpretation of Sterling_numbers_2nd_kind ; Theorem Recurrence relation for Sterling_numbers_2nd_kind;
  11. 11 pg 11: @35:54 Statement of the importance of the Sterling numbers; important question @37:23 ; suggestion to review starting WLA1_pg7 @;
  12. 12 pg 12: @40:48 Of primary importance to problems of practical application; Non_standard ideas; This is at the heart of change of basis @;
  13. 13 pg 13: @ Transpose a matrix and vector;
  14. 14 pg 14: @50:11 Application of this effect of change of basis on coordinate vectors: analyse a polynomial sequence; Newtons formula; A very useful thing to be able to do @;
  15. 15 pg 15: @ General C: transpose of signed Stirling matrix of 1st kind;
  16. 16 pg 16: @ Exercises 23.1-3;
  17. 17 pg 17: @56:13 Exercises 23.4-5; closing remarks @; THANKS to EmptySpaceEnterprise
  18. 18 Introduction
  19. 19 James Stirling Methodus Differentialis
  20. 20 Pascal Matrix
  21. 21 Combinatorial interpretation
  22. 22 Number theoretic interpretation
  23. 23 Inverting Pascal matrices
  24. 24 Inverting Stirling matrices
  25. 25 Stirling numbers of the second kind
  26. 26 Combinatorial interpretation of Stirling numbers
  27. 27 Square pyramidal numbers
  28. 28 Transpose of signed Stirling matrix of first kind

Never Stop Learning.

Get personalized course recommendations, track subjects and courses with reminders, and more.

Someone learning on their laptop while sitting on the floor.