# Some Symmetric Identities Involving Fubini Polynomials and Euler Numbers

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

**Corollary**

**1.**

**Corollary**

**2.**

**Corollary**

**3.**

**Corollary**

**4.**

**Corollary**

**5.**

**Corollary**

**6.**

## 2. A Simple Lemma

**Lemma**

**1.**

**Proof.**

## 3. Proof of the Theorem

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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$\mathit{C}(\mathit{k},\mathit{i})$ | $\mathit{i}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0$ | $\mathit{i}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}1$ | $\mathit{i}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}2$ | $\mathit{i}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}3$ | $\mathit{i}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}4$ | $\mathit{i}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}5$ | $\mathit{i}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}6$ | $\mathit{i}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}7$ | $\mathit{i}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}8$ | $\mathit{i}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}9$ |
---|---|---|---|---|---|---|---|---|---|---|

$k\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}1$ | 1 | 1 | ||||||||

$k\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}2$ | 1 | 3 | 2 | |||||||

$k\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}3$ | 1 | 6 | 11 | 6 | ||||||

$k\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}4$ | 1 | 10 | 35 | 50 | 24 | |||||

$k\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}5$ | 1 | 15 | 85 | 225 | 274 | 120 | ||||

$k\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}6$ | 1 | 21 | 175 | 735 | 1624 | 1764 | 720 | |||

$k\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}7$ | 1 | 28 | 322 | 1960 | 6769 | 13,132 | 13,068 | 5040 | ||

$k\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}8$ | 1 | 36 | 546 | 4536 | 22,449 | 67,284 | 118,124 | 109,584 | 40,320 | |

$k\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}9$ | 1 | 45 | 870 | 9450 | 63,273 | 269,325 | 723,680 | 1,172,700 | 1,026,576 | 362,880 |

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**MDPI and ACS Style**

Jianhong, Z.; Zhuoyu, C.
Some Symmetric Identities Involving Fubini Polynomials and Euler Numbers. *Symmetry* **2018**, *10*, 303.
https://doi.org/10.3390/sym10080303

**AMA Style**

Jianhong Z, Zhuoyu C.
Some Symmetric Identities Involving Fubini Polynomials and Euler Numbers. *Symmetry*. 2018; 10(8):303.
https://doi.org/10.3390/sym10080303

**Chicago/Turabian Style**

Jianhong, Zhao, and Chen Zhuoyu.
2018. "Some Symmetric Identities Involving Fubini Polynomials and Euler Numbers" *Symmetry* 10, no. 8: 303.
https://doi.org/10.3390/sym10080303