Editing Stirling number (section)

== Similar properties ==
{| class="wikitable"
|+Table of similarities
![[Stirling numbers of the first kind]]
![[Stirling numbers of the second kind]]
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|<math> \left[{n+1\atop k}\right] = n \left[{n\atop k}\right] + \left[{n\atop k-1}\right]</math>
|<math>\left\{{n+1\atop k}\right\} = k \left\{{ n \atop k }\right\} + \left\{{n\atop k-1}\right\} </math>
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|<math>\sum_{k=0}^n \left[{n\atop k}\right] = n!</math>
|<math>\sum_{k=0}^n \left\{ {n \atop k} \right\} = B_n</math>, where <math>B_n</math> is the <math>n</math>th [[Bell number]]
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|<math>\sum_{k=0}^n \left[{n\atop k}\right] x^k = x^{(n)}</math>, where <math>\{x^{(n)}\}_{n\in\N} </math> are the [[Falling and rising factorials|rising factorials]]
|<math>\sum_{k=0}^n \left\{ {n \atop k} \right\} x^k = T_n(x)</math>, where <math>\{T_n\}_{n\in\N} </math> are the [[Touchard polynomials]]
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|<math> \left[{n\atop 0}\right] = \delta_n,\ \left[{n\atop n-1}\right] = \binom{n}{2},\ \left[{n\atop n}\right] = 1</math>
|<math> \left\{{n\atop 0}\right\} = \delta_n,\ \left\{{n\atop n-1}\right\} = \binom{n}{2},\ \left\{{n\atop n}\right\} = 1</math>
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|<math>\left[{n+1\atop k+1}\right] = \sum_{j=k}^n \left[{n\atop j}\right] \binom{j}{k}  </math>
|<math>\left\{{n+1\atop k+1}\right\} = \sum_{j=k}^n \binom{n}{j} \left\{{ j \atop k }\right\} </math>
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|<math>\left[\begin{matrix} n+1 \\ k+1 \end{matrix} \right] = \sum_{j=k}^n \frac{n!}{j!} \left[{j \atop k} \right]</math>
|<math>
\left\{{n+1\atop k+1}\right\} = \sum_{j=k}^n (k+1)^{n-j} \left\{{j \atop k}\right\} </math>
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|<math>\left[{ n+k+1 \atop n }\right] = \sum_{j=0}^k (n+j) \left[{n+j \atop j}\right]</math>
|<math>\left\{{n+k+1 \atop k}\right\} = \sum_{j=0}^k j \left\{{ n+j \atop j }\right\}</math>
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|<math>\left[{n \atop l+m } \right] \binom{l+m}{l} = \sum_k \left[{k \atop l} \right] \left[{n-k \atop m } \right] \binom{n}{k} </math>
|<math>\left\{{n \atop l+m } \right\} \binom{l+m}{l} = \sum_k \left\{{k \atop l} \right\} \left\{{n-k \atop m } \right\} \binom{n}{k} </math>
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|<math>\left[{n+k \atop n} \right] \underset{n \to \infty}{\sim} \frac{n^{2k}}{2^k k!}. </math>
|<math> \left\{{n+k \atop n}\right\} \underset{n \to \infty}{\sim} \frac{n^{2k}}{2^k k!}.</math>
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|<math>\sum_{n=k}^\infty\left[{n\atop k}\right] \frac{x^n}{n!} = \frac{(-\log(1-x))^k}{k!}.</math>
|<math> \sum_{n=k}^\infty \left\{ {n \atop k} \right\} \frac{x^n}{n!} = \frac{(e^x-1)^k}{k!}.</math>
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|<math> \left[{n \atop k} \right] = \sum_{0 \leq i_1 < \ldots < i_{n-k} < n} i_1 i_2 \cdots i_{n-k}.</math>
|<math>
\left\{ {n \atop k} \right\} = \sum_{
\begin{array}{c}
c_1 + \ldots + c_k = n-k\\
c_1, \ldots,\ c_k\ \geq\ 0
\end{array}
} 1^{c_1} 2^{c_2} \cdots k^{c_k} 
</math>
|}
See the specific articles for details.