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Pascal Matrix
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Stirling Numbers and Pascal Triangles - Foundations of Linear Algebra
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- 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 Stirli…
- 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 re…
- 3 pg 3: @ Calculating Stirling numbers; Theorem Recurrence relation: Stirling numbers; proof;
- 4 pg 4: @ Pascal's triangle and binomial coefficients; recurrence relation for binomial coefficients; Pascal matrix;
- 5 pg 5: @ Combinatorial interpretation of Sterling numbers;
- 6 pg 6: @17:34 Number theoretic interpretation of Sterling numbers; summary of Sterling number interpretation @;
- 7 pg 7: @ Sterling numbers of the 2nd kind; Inverting the Pascal matrices;
- 8 pg 8: @26:36 Inverting Stirling matrices; reintroduction of some ignored symmetry @ ; Sterling matrix of the 2nd kind;
- 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 pg 10: @ Combinatorial interpretation of Sterling_numbers_2nd_kind ; Theorem Recurrence relation for Sterling_numbers_2nd_kind;
- 11 pg 11: @35:54 Statement of the importance of the Sterling numbers; important question @37:23 ; suggestion to review starting WLA1_pg7 @;
- 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 pg 13: @ Transpose a matrix and vector;
- 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 pg 15: @ General C: transpose of signed Stirling matrix of 1st kind;
- 16 pg 16: @ Exercises 23.1-3;
- 17 pg 17: @56:13 Exercises 23.4-5; closing remarks @; THANKS to EmptySpaceEnterprise
- 18 Introduction
- 19 James Stirling Methodus Differentialis
- 20 Pascal Matrix
- 21 Combinatorial interpretation
- 22 Number theoretic interpretation
- 23 Inverting Pascal matrices
- 24 Inverting Stirling matrices
- 25 Stirling numbers of the second kind
- 26 Combinatorial interpretation of Stirling numbers
- 27 Square pyramidal numbers
- 28 Transpose of signed Stirling matrix of first kind
