Finding a paper/reference where a solution also exists I have already calculated the following sum:
$$\sum_{k=0}^\infty{B_2(k)}=\frac{\pi^2}{3}-2\approx1.2898681336964$$
where $B_2(k)$ is the 2nd hypergeometric Bernoulli number of order $k$.  These numbers are defined with the following generating function:
$$\frac{x^N/N!}{e^x-1-x-...-(x^{N-1}/(N-1)!)}=\sum_{k=0}^\infty\frac{B_k(N)}{k!}x^k$$
My question is this:  Is there a way to find a catalog of numbers such as this to see if this number is the result of other calculations, summations, products, integrals, etc?  I checked OEIS for the decimal sequence but is was not there but didn't know if other such sites/references existed.
While the below answer is helpful for another way to get the closed form of the sum, I do not need that.  Hopefully someone else has the answer I am looking for.  Thanks.

Observe that we are trying to compute the sum
$$2\sum_{k=0}^\infty 
[x^2] \frac{x^k/k!}{e^x-1-x-\cdots-x^{k-1}/(k-1)!}$$
which is
$$2\sum_{k=0}^\infty 
[x^2] \frac{x^k/k!}{\sum_{q=k} \frac{x^q}{q!}}.$$
Use the Cauchy Product to extract the coefficient:
$$\frac{x^k}{k!}
= \left(\sum_{q\ge k} \frac{x^q}{q!}\right)
\times\left(a_0 + a_1 x + a_2 x^2 + \cdots\right).$$
This gives $a_0 =1$ and
$$a_0 \frac{1}{(k+1)!} + a_1 \frac{1}{k!} = 0 $$
so that $a_1 = - \frac{1}{k+1}.$
Furthermore
$$a_0 \frac{1}{(k+2)!} + a_1 \frac{1}{(k+1)!} 
+ a_2 \frac{1}{k!} = 0$$
or
$$a_0  + a_1 (k+2)
+ a_2 (k+1)(k+2) = 0$$
so that $a_2 = \frac{1}{(k+1)^2(k+2)}.$

Hence we are computing the sum
$$\sum_{k\ge 0} \frac{1}{(k+1)^2(k+2)}$$
which is by partial fractions
$$\sum_{k\ge 0}
\left(\frac{1}{k+2}-\frac{1}{k+1} +\frac{1}{(k+1)^2}\right).$$
which evaluates to 
$$-1 + \zeta(2).$$
Multiply by two to get
$$2\zeta(2) - 2 = \frac{\pi^2}{3} - 2.$$
