Problem 45 - Calculus Single Variable

Question

Recursively define two sequences \(\left\\{c_{n}\right\\}_{n=1}^{\infty}\) and \(\left\\{C_{n}\right\\}_{n=1}^{\infty}\) by initializing \(c_{0}=C_{0}=1\) and, for each nonnegative integer \(n,\) setting $$ c_{n+1}=c_{0} c_{n}+c_{1} c_{n-1}+\cdots+c_{n-1} c_{1}+c_{n} c_{0} $$ and $$ C_{n+1}=\frac{4 n+2}{n+2} C_{n} $$ Calculate \(c_{n}\) and \(C_{n}\) for \(0 \leq \mathrm{n} \leq 8 .\) (These integer sequences can be shown to be the same for all nonnegative integers \(n .\) The members of these sequences are called Catalan numbers.) If \(P(x)\) is an assertion about \(x\) that is either true or false depending on the value of \(x,\) then we can make \(P(x)\) into a numerical function by using 1 for true and 0 for false. Thus the function \(x \mapsto\left(x^{2}

Step-by-Step Solution

Answer

The Catalan numbers for \(0 \leq n \leq 8\) are: 1, 1, 2, 5, 14, 42, 132, 429, and 1430.

Step 1: Initialize the Sequences

Start with the initial values, where both sequences are initialized as follows: \( c_0 = 1 \) and \( C_0 = 1 \). These will be the base cases for our recursive definitions.

Step 2: Calculate Catalan Numbers (Recursive Calculation for \(c_n\))

Use the given recursion formula for \(c_n\): \[ c_{n+1} = c_0 c_n + c_1 c_{n-1} + \cdots + c_{n-1} c_1 + c_n c_0 \] Starting with \(c_0 = 1\), calculate subsequent values:- For \(c_1\): \( c_1 = c_0 c_0 = 1 \)- For \(c_2\): \( c_2 = c_0 c_1 + c_1 c_0 = 1 + 1 = 2 \)- For \(c_3\): \( c_3 = c_0 c_2 + c_1 c_1 + c_2 c_0 = 2 + 1 + 2 = 5 \)- For \(c_4\): \( c_4 = c_0 c_3 + c_1 c_2 + c_2 c_1 + c_3 c_0 = 5 + 2 + 2 + 5 = 14 \)- Continue this pattern to calculate up to \(c_8\).

Step 3: Calculate Catalan Numbers (Recursive Calculation for \(C_n\))

Use the given recursion formula for \(C_n\): \[ C_{n+1} = \frac{4n+2}{n+2} C_n \] Start from \(C_0 = 1\) and calculate subsequent values:- For \(C_1\): \( C_1 = \frac{4(0)+2}{0+2} C_0 = 1 \)- For \(C_2\): \( C_2 = \frac{4(1)+2}{1+2} C_1 = 2 \)- For \(C_3\): \( C_3 = \frac{4(2)+2}{2+2} C_2 = 5 \)- For \(C_4\): \( C_4 = \frac{4(3)+2}{3+2} C_3 = 14 \)- Continue this pattern to calculate up to \(C_8\).

Step 4: Verify the Sequences are the Same

Observe that both sequences \( \{c_n\} \) and \( \{C_n\} \) produce the same values for each \( n \) from the calculations in Steps 2 and 3. This verifies the given claim that these sequences, defined recursively, are equivalent for Catalan numbers.

Key Concepts

Recursive Sequences

Recursive sequences are mathematical tools used to define a sequence of numbers where each term is derived based on previous terms. In this exercise, two recursive sequences are given:

• The first sequence, \(\{c_n\}_{n=0}^{\infty}\), uses the formula \(c_{n+1} = c_0 c_n + c_1 c_{n-1} + \cdots + c_{n} c_0\). Initially, you start with \(c_0 = 1\), and each subsequent term is calculated using the sum of product pairs of earlier terms.
• The second sequence, \(\{C_n\}_{n=0}^{\infty}\), follows the formula \(C_{n+1} = \frac{4n+2}{n+2} C_n\), again beginning with \(C_0 = 1\). Each term in this sequence scales the previous term by a fraction that depends on the current index \(n\).

Recursive sequences are particularly powerful in defining Catalan numbers, which are encountered in a wide range of combinatorial mathematics problems such as counting certain types of lattice paths and parenthetical expressions.

Mathematical Induction

Mathematical induction is a proof technique that verifies a statement is true for all natural numbers. It consists of two primary steps:

• Base Case: Prove the statement is true for the initial value (usually \(n=0\) or \(n=1\)). For these sequences, the base cases were given as \(c_0 = 1\) and \(C_0 = 1\).
• Inductive Step: Show that if the statement holds for some arbitrary natural number \(k\), then it must also hold for \(k + 1\). For instance, given \(c_k\), show how you can determine \(c_{k+1}\) using its recursive formula.

Induction is a valuable method in recursively proving statements about sequences like Catalan numbers, ensuring that each calculated step is built upon a former verified calculation.

Piecewise Functions

Piecewise functions are mathematical functions defined by multiple sub-functions, each applying to a specific interval of the domain. These functions can be expressed with a single formula by making use of logical conditions:

• For the function \(F(x)\) given in this exercise, the value of \(x\) determines which expression to use:
  • \(F(x) = -x + 3\) when \(x < 0\)
  • \(F(x) = 3\) when \(0 < x < 2\)
  • \(F(x) = x + 1\) when \(x \geq 2\)
• Using logical operations, this can be expressed as a single formula: \(F(x) = -x \cdot (x<0) + 3 \cdot (0

Piecewise functions allow complex functions to be neatly encapsulated and evaluated for any given input, often simplifying the handling of functions using computational methods.

Truth Value Assignment

Truth value assignment is the process of converting logical statements into numerical values to perform calculations. This technique, often used in programming and mathematics, simplifies the handling of piecewise cases.

• When a logical condition is true, it is assigned a value of 1. If false, it turns to 0. Thus, these can be used as multipliers in functions to efficiently select the appropriate sub-expression without branching the calculation.
• For example, with the function \(F(x)\) presented, each condition checks whether \(x\) satisfies a particular constraint, then uses its boolean output multiplied by a function.

This technique of truth value assignment eases the coding and calculation of piecewise functions, providing a streamlined approach to dealing with multiple conditions in both theoretical and applied contexts.