Let $c_0,c_1,c_2,\dots$ be the sequence defined so...

Formulating the problem

I’m mapping out the expression $\frac{1 - 3x - \sqrt{1 - 14x + 9x^2}}{4}$ as a generating function for the sequence $c_k$. The matrix $A$ has a Toeplitz structure, and I’m aiming to determine its determinant.

Examine relationships

I’m noting the known relationship between the generating function and the determinant of a Toeplitz matrix. Factoring the expression under the square root is next.

Identifying key factors

I’m working through the discriminant for $t^2 - 14t + 9 = 0$. This leads to solutions $a = 7 + 2\sqrt{10}$ and $b = 7 - 2\sqrt{10}$.

Breaking down the expression

I'm thinking through rewriting $\frac{1 - 3x - \sqrt{(1 - \alpha x)(1 - \beta x)}}{4}$ in a way that reveals its connection to the generating function.

Pondering strategies

I'm exploring binomial expansion or partial fractions to find a simpler form for the generating function, aiming to express $c_k$ in a closed form or use a generalized Chebyshev polynomial approach.

Analyzing coefficients

I’m working through the series expansion of $\sqrt{1 - 14x + 9x^2}$ and examining if $c_k$ fits a linear recurrence. This might lead to identifying a closed form for the Toeplitz determinant.

Confirming the expansion

I’m confirming the expansion of $\sqrt{1 - 14x + 9x^2}$ and suggesting a rational combination approach to identify potential patterns or recurrence relations for $c_k$.

Reworking the equation

Expanding the terms to address the $\sqrt{1 - 14x + 9x^2}$ expression, I might identify patterns or recurrence relations for $c_k$.

Isolating terms

I’m rearranging the equation to isolate $\sqrt{1 - 14x + 9x^2}$, ultimately confirming the expression simplifies to $8x$. This step is crucial for further simplification.

Isolating again

I’m reiterating the isolation of $\sqrt{1 - 14x + 9x^2}$ by equating expressions, leading to the form $\frac{x - 2 G(x)^2}{G(x)}$ and verifying with the original definition.

Equating expressions

By isolating terms and equating expressions, I find that $G(x)$ satisfies a quadratic equation in terms of $x$. This leads to further insights into the relationship between the terms.

Examining coefficients

I’m analyzing the coefficients of $G(x)$ by leveraging convolution and linear recurrence relations. This approach highlights the interplay between $c_{i+j-1}$ in the matrix.

Aligning coefficients

I'm aligning coefficients in the expression, noting that for $n=0$, only $c_0$ contributes, resulting in a value of 0.

Analyzing coefficient patterns

OK, let's break down the coefficients. For $n=1$, $c_1$ equals 1. For $n \geq 2$, $c_2$ is found using the recurrence relation, leading to $c_2 = 5$.

Calculating coefficients

I’m piecing together the coefficients $c_3$ and $c_4$ using the recurrence relation. The ratios don't suggest a straightforward pattern, so I’ll investigate further.

Solving for G(x)

I’m working through the quadratic equation for $G(x)$ and pinpointing the correct sign for the solution. Calculating the discriminant helps in identifying valid roots.

Expressing G(x)

I’m working through expressing $G(x)$ using binomial expansions for $(1 - \alpha x)^{1/2}$ and $(1 - \beta x)^{1/2}$, ensuring the sums are valid.

Generating new insights

I’m working through the sums and noticing the coefficients don't fit known integer sequences. This suggests the coefficients might be special or have a non-integer ratio.

Piecing together patterns

I’m looking at the differences in the sequence and detailing the sums for $d_n$, noting that $c_0$ being zero simplifies the calculations.

Revisiting the recursion

I’m working through the sum $\sum_{k=1}^{n-1} c_k c_{n-k}$, separating $c_{n-1}$ and linking it to $d_{n-1}$ to refine $c_n$'s expression.

Validating the formula

I’m confirming the formula for $c_n$ by plugging in known values and ensuring the recursion is consistent, especially leveraging the symmetry of the sums for validation.

Polishing the approach

OK, let me see. I’m refining the recurrence by isolating terms and exploring if $c_n$ fits a homogeneous linear recurrence of order 3 or 4, considering the generating function's complexity.

Breaking down the expression

I’m trying to reframe the expression $\frac{1 - 3x - \sqrt{1 - 14x + 9x^2}}{4}$ in terms of $\alpha$ and $\beta$, and seeing if it forms a homogeneous linear recurrence.

Crunching specifics

I’m mapping out how the determinant of a Toeplitz matrix, with a symbol tied to the generating function, involves its factorization. This leads to considering formulas for the determinant.

Evaluating determinant patterns

I’m gathering info on any consistent patterns in the determinants for small Toeplitz matrices, specifically focusing on matrices of dimensions 1 and 2.

Calculating determinants

I’m wrapping up the determinants for small Toeplitz matrices by expanding $n \times n$ matrices, specifically working through the steps to compute the determinant for a 3x3 matrix.

Crunching the numbers

OK, let's break this down: calculating minors for the 3x3 matrix to determine a determinant of 1000, with smaller matrices showing a pattern in determinants for $n=1$ and $n=2$.

Taking a closer look

I'm mapping out the pattern in determinants: each time, the determinant increases significantly, suggesting a formula like $10^{n(n-1)/2}$ might apply. This hypothesis needs further testing.

Piecing together

I’m working through the idea that the determinant formula for a $n \times n$ Toeplitz matrix might be $10^{n(n-1)/2}$. This pattern suggests a significant increase for each $n$.

Wrapping up the matrix

I’ve calculated $c_7$ and assembled the 4x4 matrix. Now, I’m thinking about computing the determinant systematically to ensure it aligns with the expected value.

Analyzing determinant approaches

I’m exploring strategies to calculate the determinant of the 4x4 matrix, including partial cofactor expansion and examining the ratio of consecutive principal minors, to ensure it aligns with the expected value.

Testing the pattern

OK, let me see. I'm examining the determinant pattern for increasing Toeplitz matrices, noticing the ratio of consecutive principal minors aligns with $10^{n(n-1)/2}$. This pattern is quite intriguing.