Taylor for Tan

Random number: 68

2nd January 2017

Back from theme parks new years florida train happy! Today, whilst working a math problem, I thought about the taylor series for $\text{tan}(x)$, more specifically the taylor series around $x = 0$, otherwise known as the Mclaurin series. By repeatedly taking the derivative of $\text{tan}(x)$, I figured out that the infinite matrix $A$ with entries $a_{i,j}$ defined such that $$a_{i,j} := \begin{cases} 0 \qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad \ \ \ \ \text{ if }\ i = 0,j \neq 1 \\ 1 \ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \ \ \ \text{ if } \ i = 0, j = 1 \\ (j-1)*a_{(i-1),(j-1)} + (j+1)*a_{(i-1),(j+1)} \ \text{ if } \ i > 0, j > 0 \\ a_{(i-1),(j+1)} \ \qquad\qquad\qquad\qquad\qquad\qquad \ \ \ \ \ \text{ if } \ i > 0, j = 0 \end{cases}$$ has the property that $$\text{tan}(x) \approx \sum_{n = 0}^{\infty} a_{n,0}\frac{x^n}{n!}$$ i.e. is equivalent to the coefficeints for the taylor series of $\text{tan}(x)$. Roughly, for the first few values we have that: $$\text{tan}(x) \approx x + \frac{2x^3}{3!} + \frac{16x^5}{5!} + \frac{272x^7}{7!} + ...$$ After digging around some more, I found out that these coefficients also go by the name of the 'Tangent Numbers' or 'Zag numbers'. Define $Z_n$ to be the number of permutations of $n$ numbers such that the sequence alternates, e.g. that there does not exist numbers $a,b,c$ in the permutation such that $a < b < c$ or $a > b > c$. Now define $A_n$ to be $Z_n / 2$. It is curious that $$\text{tan}(x) \approx A_1 x + A_3 \frac{x^3}{3!} + A_5 \frac{x^5}{5!} + ... $$ I'll try and figure out the relationship between the number of alternating permutations and the infinite matrix $A$ later.