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I'm trying to figure out the rule that's producing the series below. The first column is the factorials $1!,2!,3!,4!,5!,6!$ But I can't figure out what else is going on. So I know the next row will start with $7!=5040$, but can't say much beyond that. If someone has insight as to what the next numbers in the series are that would be much appreciated.

$1$

$2\qquad2$

$6\qquad12\qquad6$

$24\qquad72\qquad72\qquad24$

$120\qquad480\qquad720\qquad480\qquad120$

$720\qquad3600\qquad7200\qquad7200\qquad3600\qquad720$

Thanks

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Have you tried dividing each row by the first term? –  Qiaochu Yuan Jan 3 '12 at 7:08
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oeis.org/… –  sdcvvc Jan 3 '12 at 7:08
    
@QiaochuYuan Commented while I was posting my answer, d'oh! –  Steven Stadnicki Jan 3 '12 at 7:12
    
@sdcvvc what a cool website –  ben Jan 3 '12 at 7:29
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1 Answer

up vote 5 down vote accepted

The easiest way to find the answer here is to divide out by the first terms, producing:

$$\begin{array} &&&&&&1\\ &&&&1&&1\\ &&&1&&2&&1\\ &&1&&3&&3&&1\\ &1&&4&&6&&4&&1\\ \end{array}$$ ...which should look awfully familiar. This implies that the $(n,k)$ entry of your series is $n! {n-1 \choose k}$.

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Thanks! Much clearer now. Pardon my ignorance but I am not familiar with the notation you used. What exactly does $n! {n-1 \choose k}$ mean? –  ben Jan 3 '12 at 7:32
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Well, the $n!$ piece of that you already know; $n-1\choose k$ (generally pronounced '$n$-minus-1 choose $k$') is the Binomial Coefficient. $a\choose b$ is defined as $a!/b!(a-b)!$ and represents the number of ways of choosing b items from a population of a items; they're arguably the most fundamental objects in combinatorics. See en.wikipedia.org/wiki/Binomial_coefficient for more details on it! –  Steven Stadnicki Jan 3 '12 at 7:56
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