6.5. A Summation Method Asume that:

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MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 30, Wednesday, November 12

6.5. A Summation Method Asume that: A(x) = S anxn, B(x) = S bnxn, and C(x) = S cnxn. If bn = dan, then B(x) = dA(x). If cn = an + bn, then C(x) = A(x) + B(x). If cn = a0bn + a1bn-1 + ... + an-1b1 + anb0, then C(x) = A(x)B(x). If bn = an-k, exept bi = 0 for i < k, then B(x) = xk A(x).

Example 1 Build a generating function h(x) for ar = 2r2. Answer: h(x) = 2x(1 + x)/(1 – x)3.

Example 2 Build a generating function h(x) with ar = (r + 1)/(r – 1) Answer: h(x) = 6x2(1 – x)-4.

Theorem 1 If h(x) is a generating function for ar, then h*(x) = h(x)/(1 – x) is the generating function for the sums of the ars. That is, h*(x) = a0 + (a0 + a1)x + (a0 + a1 + a2)x2 + ...

Example 1 (continued) Evaluate the sum: h*(x) = h(x)/(1 – x) = 2x/(1 - x)4 + 2x2/(1 – x)4. Answer: 2C(n+2,3) + 2C(n + 1, 3).

Example 2 (continued) Evaluate the sum 3 £ 2 £ 1 + 4 £ 3 £ 2 + ... + (n+1)n(n-1). Answer:h*(x) = 6x2/(1 – x)5. The coefficient at xn-2 is 6C(n + 2, 4).