Find a formula for counting the number of sequences where the product of absolute value terms is bounded by a given number | Step-by-Step Solution

Hi there! I can see you're working on an interesting counting problem. Let me help you break this down step by step!

What We're Solving:

We need to count how many sequences (a₀, a₁, a₂, ..., aₖ) exist such that |a₀| × |a₁| × |a₂| × ... × |aₖ| ≤ n, where n is our given bound.

The Approach:

This is a beautiful problem that combines number theory with counting! We're essentially asking: "In how many ways can we write a product of absolute values that doesn't exceed n?" The key insight is that we only care about the absolute values, so we can think of this as counting sequences of positive integers first, then accounting for signs later.

Step-by-Step Solution:

Step 1: Simplify by focusing on absolute values Since we only care about |aᵢ|, let's first count sequences of positive integers (b₀, b₁, ..., bₖ) where b₀ × b₁ × ... × bₖ ≤ n.

Step 2: Connect to number theory This is equivalent to counting the number of ways to write divisors of numbers ≤ n as products of exactly (k+1) factors.

Step 3: Use a generating function approach For each position i, aᵢ can be any integer where |aᵢ| contributes to our product. The generating function for one position would be: ∑(all possible absolute values) x^(absolute value)

Step 4: Consider the constraint systematically We need to count ordered sequences where:

• Each |aᵢ| ≥ 1 (assuming we want non-zero terms)
• The product |a₀| × |a₁| × ... × |aₖ| ≤ n

Step 5: Account for signs Once we count the valid absolute value sequences, each aᵢ can be positive or negative, giving us 2^(k+1) sign combinations for each valid absolute value sequence.

The Answer:

The formula is: 2^(k+1) × N(k,n)

Where N(k,n) = number of ways to choose (k+1) positive integers whose product is ≤ n.

More specifically: N(k,n) = ∑(d=1 to n) [number of ways to write d as a product of exactly (k+1) positive integers]

This can be computed using the divisor function and dynamic programming techniques.

Memory Tip:

Think "Product → Divisors → Signs"!

• 1. Product: We're constraining a product
• 2. Divisors: This connects to counting divisors and factorizations
• 3. Signs: Don't forget to multiply by 2^(k+1) for all possible sign combinations!

Remember, problems involving products of integers often connect beautifully to number theory concepts like divisors and prime factorizations. You're doing great tackling such an advanced topic! 🌟