MathDB
2017 BMT Analysis #9

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March 10, 2024
number theoryalgebra

Problem Statement

Let ada_d be the number of non-negative integer solutions (a,b)(a, b) to a+b=da + b = d where a≡ba \equiv b (mod nn) for a fixed n∈Z+n \in Z^+. Consider the generating function M(t)=a0+a1t+a2t2+...M(t) = a_0 + a_1t + a_2t^2 + ... Consider P(n)=lim⁡t→1(nM(t)−1(1−t)2).P(n) = \lim_{t\to 1} \left( nM(t) - \frac{1}{(1 - t)^2} \right). Then P(n)P(n), n∈Z+n \in Z^+ is a polynomial in nn, so we can extend its domain to include all real numbers while having it remain a polynomial. Find P(0)P(0).
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