Real Estate in Toney, AL

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30 Bayfront Lane, Scottsboro

30 Bayfront Lane, Scottsboro

Hampton Cove

Just cleaned up Hampton Cove

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

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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

Real Estate Intelligence Agency, Inc.

Looks like Mikko Jetsu and I of Marketing Intelligence will be adding IDX and MLS to Real Estate Intelligence.

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}