The function

isn't defined at *x=0*, but this singularity is removable because the limit
as x→0 is finite, as we can easily see by applying De L'Hôpital's rule.
The same thing is true for its derivatives:

We can therefore expand this function using the MacLaurin series expansion

where *B _{n}* is the limit as

In this way we can find:

and so on. The *B _{n}* are said

The calculation of the derivatives and of their limits, even using powerful software,
is slow and does'nt give any hints to conjecture a general relation between the numbers
*B _{n}* which could allow a more direct algorithm to generate these numbers.
Such algorithm can be found using the well known series expansion of the natural
exponential function in the following way.

From we have

Now we multiplicate all the addends in the second sum by every addend of the first

The first and the second term are identical only if, in the second term,
*B _{0}=1* and the sum of the coefficients of the same power of

The denominators in these sums coincide with the denominators of the factorial expansions of the binomial coefficients.

Using the binomial coefficients, we can write the n-th calculated row as

If we consider, for example, the last calculated row, we have

So, in general, we can write

This allows the recursive calculation of every *B _{n}*

To obtain further *B _{n}*, we may automatize the calculation
as in the following Javascript application.
If your browser does not allow the

`iframe`

tag, you can open the source page.From the explicitly calculated values and using the applet, we can conjecture that all the Barnoulli's numbers with odd index ≥ 3 are null. This is true, because from

we have

and after

So, every *B* in the first term must be 0.

Given the function

let *E _{n}* be the values taken by it and its successive derivatives for

and then

If we explicitly represent some of the summands of the factors in the right-hand side, we obtain

The expansion of this product is

then

Two polynomials in the same variable are identical only if the same powers of the variable have equal coefficients,
so the coefficients of powers of *x* in the right-hand side must coincide with the coefficients of the series
expansion of the exponential. These identities allow us to calculate recursively the values of *E _{n}*. Infact

Obviously *E _{5}*, like

We can conjecture that, in general,

To obtain further *E _{n}*, we may automatize the calculation
as in the following Javascript application.
If your browser does not allow the

`iframe`

tag, you can open the source page.There are different conventions on the notation of Euler numbers. Here we assume the convention adopted by *Mathematica*
for the function `EulerE[n]`

.