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Is there any known formula for multiplying out factorials?

$x$ being a positive integer, $x!$ is defined as $$x(x-1)(x-2)\cdots3\cdot2\cdot1$$

My question is if there exists another "multiplied out" form of that product, as a sum.

I tried multiplying out the first three, four, five factors and seemingly random coefficients that seemed to get bigger and bigger the more factors I multiplied out appeared.

Blue
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Iridium
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    This might help: https://math.stackexchange.com/questions/1343452/any-shortcut-to-calculate-factorial-of-a-number-without-calculator-or-n-to-1 – I am a person May 08 '21 at 18:03
  • Please be sure you format the question using mathjax. https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference – Phicar May 08 '21 at 18:29

1 Answers1

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Yes, indeed. This has to do with something called Stirling numbers of the first kind. The numbers you are looking at are $$n!=n(n-1)\cdots (n-n+1)=\sum _{k=0}^n(-1)^{n-k}{n\brack k}n^k,$$ where ${n\brack k}$ is the number of permutations of $n$ having $k$ cycles.

Phicar
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