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Question: Given the sequence of numbers:$$\frac 16,\frac 1{90},\frac 1{945},\frac 1{9450},\frac 1{93555},\frac {691}{638512875},\frac 2{18243225},\cdots$$ What would be a sequence that generates these numbers?

I'm trying to generalize a summation problem, and for me to continue, I need to find a general sequence that you can use to generate the above numbers.

I don't know where to begin with this kind of problem. I've never really encounter these kinds of problems. And I'm not sure that even Wolfram Alpha can solve for a sequence. Note that this is not a homework problem.

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    Have you tried $\large\frac{\zeta(2n)}{\pi^{2n}}$? These numbers are the coefficients of the Taylor series of $$\frac{1 - z \cot z}{2 z^2}$$ at $z=0$. – 2017-02-03
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    It would help of you could tell us where this sequence comes from. Any finite sequence has a huge number of possible formulae. – 2017-02-03
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    It does look like $\zeta(2n)/\pi^{2n}=\frac{(-1)^{n+1}B_{2n}2^{2n-1}}{(2n)!}$ where $B_{n}$ are the Bernoulli numbers. See https://en.wikipedia.org/wiki/Particular_values_of_the_Riemann_zeta_function – 2017-02-03
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    https://oeis.org/A002432 – 2017-02-03
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    Yes, Euler generates these numbers in "Concerning the sums of series of reciprocals" (Comment. acad. sc. Petrop. 7. 1740, p. 24 onwards, translated by Ian Bruce) – 2017-02-03
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    Why are the answers getting downvoted? – 2017-02-03
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    @ThomasAndrews Um... what are Bernoulli numbers? Do you know where I can get an introduction to them? – 2017-02-03
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    @Frank I provided a Wikipedia link, and that article has a link for Bernoulli numbers – 2017-02-03

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