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# Closed Forms for Derangement Numbers in Terms of the Hessenberg Determinants

1. Institute of Mathematics, Henan Polytechnic University, Jiaozuo 454010, Henan~Province, China
2. College of Mathematics, Inner Mongolia University for Nationalities, Tongliao 028043, Inner Mongolia Autonomous Region, China
3. Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin 300387, China
4. Department of Mathematics and Informatics, Weinan Teachers University, Weinan 714000, Shaanxi Province, China
5. School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo~454010, Henan Province, China
Version 1 : Received: 11 October 2016 / Approved: 11 October 2016 / Online: 11 October 2016 (10:53:07 CEST)

Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite: Qi, F.; Zhao, J.; Guo, B. Closed Forms for Derangement Numbers in Terms of the Hessenberg Determinants. Preprints 2016, 2016100035 (doi: 10.20944/preprints201610.0035.v1). Qi, F.; Zhao, J.; Guo, B. Closed Forms for Derangement Numbers in Terms of the Hessenberg Determinants. Preprints 2016, 2016100035 (doi: 10.20944/preprints201610.0035.v1).

## Abstract

In the paper, the authors find closed forms for derangement numbers in terms of the Hessenberg determinants, discover a recurrence relation of derangement numbers, present a formula for any higher order derivative of the exponential generating function of derangement numbers, and compute some related Hessenberg and tridiagonal determinants.

## Subject Areas

derangement number; closed form; Hessenberg determinant; tridiagonal determinant; generating function; recurrence relation; derivative

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Comment 1
Commenter: Honest John China
Commenter's Affiliation: China University
The commenter has declared there is no conflict of interests.
Comment: Nice paper!
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