By David Guichard
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Extra resources for An Introduction to Combinatorics and Graph Theory
Show that the Maclaurin series for (x + 1)r is ∞ i=0 r i xi . 3. 4, show that all coefficients beginning with x16 are 180. 4. Use a generating function to find the number of solutions to x1 + x2 + x3 + x4 = 14, where 0 ≤ x1 ≤ 3, 2 ≤ x2 ≤ 5, 0 ≤ x3 ≤ 5, 4 ≤ x4 ≤ 6. 5. Find the generating function for the number of solutions to x1 + x2 + x3 + x4 = k, where 0 ≤ x1 ≤ ∞, 3 ≤ x2 ≤ ∞, 2 ≤ x3 ≤ 5, 1 ≤ x4 ≤ 5. 6. Find a generating function for the number of non-negative integer solutions to 3x + 2y + 7z = n.
So when i is large, √ 1 1+ 5 i Fi = round √ , 2 5 √ 5)/2| < 1, the limit of that is, the first term rounded to the nearest integer. As it turns out, this is true starting with i = 0. You can see how to do the entire solution in Sage. 4. 1. Find the generating function for the solutions to hn = 4hn−1 − 3hn−2 , h0 = 2, h1 = 5, and use it to find a formula for hn . 2. Find the generating function for the solutions to hn = 3hn−1 + 4hn−2 , h0 = h1 = 1, and use it to find a formula for hn . 3. Find the generating function for the solutions to hn = 2hn−1 + 3n , h0 = 0, and use it to find a formula for hn .
N (−1)n−j j=1 n−1 k j−1 j−1 k (−1)n−j−1 n−1 j n−1 =k j=1 n (−1)n−j (n − 1) + j=1 j k n−1 j n−1 − (n − 1) (−1)n−j−1 j=1 j k n−1 j j k = kδn−1,k − (n − 1)δn−1,k . Now if k = n − 1, this is (n − 1)δn−1,n−1 − (n − 1)δn−1,n−1 = 0, while if k < n − 1 it is kδn−1,k − (n − 1)δn−1,k = k · 0 − (n − 1) · 0 = 0. If we interpret the triangles containing the s(n, k) and S(n, k) as matrices, either m × m, by taking the first m rows and columns, or even the infinite matrices containing the entire triangles, the sums of the theorem correspond to computing the matrix product in both orders.
An Introduction to Combinatorics and Graph Theory by David Guichard