The number of non-negative integral solutions of $x+y+z \leq n$ where $n \in \mathbb{N}$ is:
Step-by-Step Solution
Key Concept: Extract recurrence relations from coefficient matching in power series, then solve using standard techniques for linear and quadratic recurrences.
Given $A_k = 4$ and expanding $(4 + B_{-1}x + C_kx^2 + ...)^2 = 4 + 4B_{-1}x + (4C_{-1} + B_{-1}^2)x^2 + ...$, establish the recurrences $B_k = 4B_{k-1} = 4^kB_1 = -4^k$ and $C_k = B_{k-1}^2 + 4C_{k-1}$. Solve the recurrence for $C_k$ using geometric series to get $C_k = 4^{2k-2} + 4^{2k-3} + ... + 4^k + 4^{k-1} = \frac{4^{k-1}(4^k-1)}{3}$.
Correct Answer: 3