Real Estate in Toney, AL

# Hampton Cove

Just cleaned up Hampton Cove

Haven’t yet figured out how to properly markup tracts (subdivisions) using Schema.

# Binomial Coefficients

$\binom{s}{t} = \prod_n \frac{t+n}{s+n}\frac{s-t+n}{n} = \prod_{n=1}^\infty \frac{t+n}{s+n}\frac{s-t+n}{n}$,

where the product is taken over the naturals, for $s \notin \{-1, -2, -3, \dots\}$, not a negative integer. Therefore,

$\binom{s}{t} = \frac{t+1}{s+1}\frac{s-t+1}{1}\frac{t+2}{s+2}\frac{s-t+2}{2}\frac{t+3}{s+3}\frac{s-t+3}{3}\cdots$

# Power of a (Ordinary) Generating Function

Consider a non zeroth cusp $(\alpha_0 \neq 0)$ (ordinary) generating function raised to a power, then
$\left(\sum_{m=0}^\infty \alpha_m x^m\right)^z = \left(\sum_m \alpha_m x^m\right)^z =\alpha_0^z\sum_{m=0}^\infty \left(\sum_{n=0}^m \frac{z^{\underline{n}}}{\alpha_0^{n}} \beta_{m,n}(\underline{\alpha})\right) x^m\\ =\sum_{m,n} \alpha_0^{z-n} z^{\underline{n}} \beta_{m,n}(\underline{\alpha}) x^m,$
where

• $m \in \mathbb{W} = \left\{0\right\} \cup \mathbb{N} = \left\{0, 1, 2, 3, \dots \right\}$
• $n \in \left\{0, 1, 2, \dots, m\right\}$
• falling factorial: $z^{\underline{1+n}}=(z-n)^{\underline{1}}z^{\underline{n}}=(z-n)z^{\underline{n}}$,

therefore,

• $z^{\underline{1}}=z$, $z^{\underline{2}}=z(z-1),\dots$
• $\left(\underline{\alpha}\right) = \left(\alpha_1,\alpha_2,\alpha_3,\dots\right)$
• $\forall \underline{\alpha}: \beta_{m,0}(\underline{\alpha})=\left[m=0\right]= \begin{cases} 1,\ m=0\,\\ 0, m \neq 0. \end{cases}$
• $n \beta_{m,n}(\underline{\alpha}) = \sum_j \beta_{j,n-1}(\underline{\alpha}) \alpha_{m-j}, j \in \left\{n-1,n,\dots,m-1\right\}$

The coefficents can be viewed as the result of a number triangle product with a column vector, \ie,

$\begin{bmatrix} 1 &0 &\cdots\\ 0 &\alpha_{1}&0&0&0&0\\ 0 &\alpha_{2}&\alpha_1^2/2!&0&0&0\\ 0 &\alpha_{3}&\alpha_{2}\,\alpha_{1}&\alpha_1^3/{3!}&0&0\\ 0 &\alpha_{4}&\alpha_3\,\alpha_1+\alpha_2^2/2&\alpha_2\,\alpha_1^2/2&\alpha_1^4/4!&0 \\ 0 &\alpha_{5} &\alpha_{4}\,\alpha_{1}+\alpha_{3} \,\alpha_{2} &\left(\alpha_2^2/2\right)\,\alpha_1+\alpha_3\,\alpha_1^2/2&\alpha_2\,\alpha_1^3/3! &\alpha_1^5/5!\\ \vdots &\vdots & & & & &\ddots \end{bmatrix} \begin{bmatrix} \alpha_0^z \\ \alpha_0^z \frac{z}{\alpha_0}\\ \alpha_0^z \frac{z(z-1)}{\alpha_0^2}\\ \alpha_0^z \frac{z(z-1)(z-2)}{\alpha_0^3}\\ \alpha_0^z \frac{z}{\alpha_0} \frac{z-1}{\alpha_0} \frac{z-2}{\alpha_0} \frac{z-3}{\alpha_0}\\ \vdots \end{bmatrix}$