Binomial Coefficients and Generating Functions - Cs.ioc.ee

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Operations on Generating Functions (6) 6. Convolution (product) If 〈f0,f1,f2,...〉 ←→ F (x) and 〈g0,g1,g2,...〉 ←→ G(x), then 〈h0,h1,h2,...〉 ←→ F (x) · G(x), where hn = f0gn + f1gn−1 + f2gn−2 + ··· + fng0. Example Hence 〈1,1,1,1...〉 · 〈0,1, 1 1 , ,...〉 = 2 3 = 〈1·0, 1·0 + 1·1, 1·0 + 1·1 + 1· 1 2 = 〈0, 1, 1 + 1 1 1 , 1 + + 2 2 3 ,···〉 = 〈0, H1, H2, H3,···〉 ∑ Hkx k0 k = 1 (1 − x) ln 1 (1 − x) . 1 1 , 1·0 + 1·1 + 1· + 1· 2 3 ,···〉
Operations on Generating Functions (7) Example 〈1,1,1,1,...〉 ←→ 1 1 − x 〈1,2,3,4,...〉 ←→ d 1 dx 1 − x = 1 (1 − x) 2 1 x 〈0,1,2,3,...〉 ←→ x · = (1 − x) 2 (1 − x) 2 〈1,4,9,16,...〉 ←→ d x 1 + x = dx (1 − x) 2 (1 − x) 3 1 + x x(1 + x) 〈0,1,4,9,...〉 ←→ x · = (1 − x) 3 (1 − x) 3
Page 1 and 2: Binomial Coefficients and Generatin
Page 3 and 4: Next section 1 Binomial coefficient
Page 5 and 6: Binomial coefficients and combinati
Page 7 and 8: Properties of Binomial Coefficients
Page 9 and 10: Yet Another Proerty 4 If n,k > 0, t
Page 11 and 12: Pascal’s Triangle 1 1 1 1 2 1 1 3
Page 13 and 14: Next section 1 Binomial coefficient
Page 15 and 16: More examples (1) 〈1,1,1,1,...〉
Page 17 and 18: More examples (2) 〈a,ab,ab 2 ,ab
Page 19 and 20: More examples (3) Taking in the las
Page 21 and 22: Next subsection 1 Binomial coeffici
Page 23 and 24: Operations on Generating Functions
Page 31: Operations on Generating Functions
Page 35 and 36: Counting with Generating Functions
Page 37 and 38: Building Generating Functions that
Page 43 and 44: How many positive integer solutions
Page 45 and 46: How many positive integer solutions
Page 47 and 48: Number of solutions of the equation
Page 49 and 50: Distribute n objects into k bins so
Page 51 and 52: Example: 100 Euro How many ways 100
Page 53 and 54: Example: 100 Euro (2) 1. Observatio
Page 55 and 56: Example: 100 Euro (3) Convolution:
Page 57 and 58: Vandermonde’s identity Binomial t
Page 59 and 60: Vandermonde’s identity (2) Equati
Page 61 and 62: Vandermonde’s identity (3) Specia
Page 63: Some other useful sequences For any

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