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I'm trying to find a general expression for the following sequence:

1, 14, 273, 7645, 296296, ...

I've already looked at the Online Encyclopedia of Integer Sequences (OEIS) and this is a known sequence, related to the Central Factorial Numbers. But they don't give a general formula for it, only a recursive one. Thanks in advance!

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    Please link to said OEIS in the future. – 2017-01-11

2 Answers 2

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In our OEIS, there is a clearly stated formula that goes as follows:

$$a(n)=s(n+3,3)^2-2s(n+3,2)s(n+3,4)+2s(n+3,1)s(n+3,5)$$

where $s(n,k)$ are the stirling numbers of the first kind.

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    Not that I want to split hairs, but these numbers themselves might not really have a closed form. – 2017-01-11
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    @ClementC. Sh... your not supposed to say that... – 2017-01-11
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    @ClementC. Well, there is this: https://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind#Other_relations – 2017-01-11
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At any case, the problem is not well defined. According to the Lagrange's Interpolation Formula, we can construct infinite sequences $a_n=p(n)$ with $p$ a polynomial and $a_1,\ldots,a_5$ the given numbers.