# In a power series, how do we keep the $\frac{1}{n!}$ term without simplification for all $x^{n}$?

I am trying to visualize the Euler numbers coming from the generating function:

\begin{align*} \sum_{n\geq0}E_{n}\frac{x^{n}}{n!} & =\text{sec}(x)+\text{tan}(x)\\ & =1+x+\frac{x^{2}}{2!}+2\frac{x^{3}}{3!}+5\frac{x^{4}}{4!}+16\frac{x^{5}}{5!}+61\frac{x^{6}}{6!}+272\frac{x^{7}}{7!}+1385\frac{x^{8}}{8!}+7936\frac{x^{9}}{9!}\cdots \end{align*}

But, when treated as usual, Series[Sec[x] + Tan[x], {x, 0, 9}] , Mathematica simplifies $$\frac{E_{n}}{n!}$$ when possible so we cannot identify coefficients neatly. Should it be enough to multiply $$n!$$, but

n = 9; Take[Range[0, n]!]*Series[Sec[x] + Tan[x], {x, 0, n}]


generates a list of $$i$$ polynomials, where only the $$i^{th}$$ term is the correct one I am looking for.

Is there a way to sieve them and thus get the desired polyomial expansion? or, Is there an easier way to keep the term $$\frac{1}{n!}$$ without simplification in the coefficient of $$x^{n}?$$

• SeriesCoefficient[Sec[x] + Tan[x], {x, 0, n}] // Simplify Mar 6 at 16:46
• Here is a similar post. Mar 6 at 21:35

If it's a question of output formatting, then perhaps this:

n = 9;
Block[{x},
x /: Power[x, k_?Positive] := Defer[x^k/k!];
Normal@ReplacePart[#,
3 :> Range[#[[4]], #[[4]] + Length@#[[3]] - 1]! #[[3]]]
] &@Series[Sec[x] + Tan[x], {x, 0, n}]

• Thank you @Michael, definitely retrieves the Euler numbers!! Just a question, this seems specially suited for the function $\text{sec}(x)+\text{tan}(x)$ but for other functions we do not get the general form $1+a_{1}x+a_{2}\frac{x^{2}}{2!}+a_{3}\frac{x^{3}}{3!}+a_{4}\frac{x^{4}}{4!}+etc.$ Consider that our even powers are generated by sec$(x)=1+\frac{x^{2}}{2!}+5\frac{x^{4}}{4!}+61\frac{x^{6}}{6!}+etc.$ whereas odd powers are generated by tan$(x)=x+2\frac{x^{3}}{3!}+16\frac{x^{5}}{5!}+etc.$ But I do not understand, why this program fails to get those series separately? Mar 6 at 18:57
• @FélixdelaFuente The 3rd part of output of Series is the list of coefficients. If the last coefficient is zero, as the degree-9 coefficient of Sec[x] is, it is truncated. The list of coefficients is shorter than Range[0, n]. I've updated it to calculate the correct length. Mar 6 at 19:08
• It works now for sec$(n)$, but somehow for tan$(x)$ it gives "almost" the correct polynomial form, but divided by the power index: $\frac{a_n\frac{x^{n}}{n!}}{n}$. Why could this happen? Mar 6 at 19:25
• @FélixdelaFuente Testing revealed it needed one more tweak. Mar 6 at 20:31
• Thank you, now it works perfecly with all Trig functions! It is nice to have them with this representation. Mar 6 at 20:44