Real Estate in Toney, AL


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,

  • 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}},


  • 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}