I believe that it is related with Stirling numbers of the second kind, but I can not find it. Please help me. Thank a lot.

Prof. Qi,

slightly related results can be found here:

http://ftp.math.uni-rostock.de/pub/romako/heft61/bou61.pdf

I saw this paper, but no my answer

I believe that one can try to find the value using the generating function.

This is an answer I proved

Moreover, we also have

where $B_{n,k}$ and $S(n,k)$ denote the partial Bell polynomials and Stirling numbers of the second kind

I found this

And this formula

where $S(n,k)$ and $s(n,k)$ are stirling numbers of the second and first kind, respectively, and $(n)_k = n(n-1)...(n-k+1)$.

This is the ref

There are many other interesting formulas in this paper derived using generating functions.

Could you share with me how did you come to your result?

it could be interesting to study is there any relation between your formula and the formula with alternating arguments of partial Bell polynomial.

My formula is essentially the same as the formula with alternating arguments of partial Bell polynomial.

Thank you very much for your recommending the nice paper for me. I am familiar with the first author of the paper, but, since I am not a combinatorialist, I did not read any of his papers.

I use these formulas for obtaining explicit formulas for computing Bernoulli numbers and Stirling numbers of the first kind.

All partial Bell polynomials with alternating arguments may be transformed into those with constant sign arguments by the following formula

This formula may be found on page 135 of the book: L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition, D. Reidel Publishing Co., Dordrecht and Boston, 1974.

Thank you, Armen Bagdasaryan

I began to learn some knowledge on Combinatorics just one month ago

Just now I sent you a new paper of mine. See you, Armen.

Thank you very much.

This formula has been derived, used, mentioned, applied in the following papers and preprints.

(1) Feng Qi and Robin J. Chapman, Two closed forms for the Bernoulli polynomials, Journal of Number Theory 159 (2016), 89--100; Available online at http://dx.doi.org/10.1016/j.jnt.2015.07.021.

(2) Feng Qi, Two closed forms for the Bernoulli polynomials, arXiv preprint (2015), available online at http://arxiv.org/abs/1506.02137.

(3) Feng Qi, An explicit formula for Bernoulli numbers in terms of Stirling numbers of the second kind, arXiv preprint (2014), available online at http://arxiv.org/abs/1401.4255.

(4) Bai-Ni Guo and Feng Qi, An explicit formula for Bernoulli numbers in terms of Stirling numbers of the second kind, Journal of Analysis & Number Theory 3 (2015), no. 1, 27--30; Available online at http://dx.doi.org/10.12785/jant/030105.

(5) Bai-Ni Guo and Feng Qi, On inequalities for the exponential and logarithmic functions and means, Malaysian Journal of Mathematical Sciences 10 (2016), no. 1, 23--34.

(6) Bai-Ni Guo and Feng Qi, On inequalities for the exponential and logarithmic functions and means, ResearchGate Technical Report (2015), available online at http://dx.doi.org/10.13140/RG.2.1.5126.2885.

(7) Feng Qi, Diagonal recurrence relations, inequalities, and monotonicity related to the Stirling numbers of the second kind, Mathematical Inequalities & Applications 19 (2016), no. 1, 313--323; Available online at http://dx.doi.org/10.7153/mia-19-23.

(8) Bai-Ni Guo and Feng Qi, Explicit formulas for special values of the Bell polynomials of the second kind and the Euler numbers, ResearchGate Technical Report (2015), available online at http://dx.doi.org/10.13140/2.1.3794.8808.

(9) Feng Qi and Bai-Ni Guo, A closed form for the Stirling polynomials in terms of the Stirling numbers, Preprints 2017, 2017030055, 4 pages; Available online at http://dx.doi.org/10.20944/preprints201703.0055.v1.

The following formally published papers are related to this question:

[1] Feng Qi, Diagonal recurrence relations for the Stirling numbers of the first kind, Contributions to Discrete Mathematics 11 (2016), no. 1, 22--30; available online at https://doi.org/10.11575/cdm.v11i1.62389

[2] Feng Qi and Jiao-Lian Zhao, Some properties of the Bernoulli numbers of the second kind and their generating function, Bulletin of the Korean Mathematical Society 55 (2018), no. 6, 1909--1920; available online at https://doi.org/10.4134/bkms.b180039

[3] Feng Qi and Bai-Ni Guo, A diagonal recurrence relation for the Stirling numbers of the first kind, Applicable Analysis and Discrete Mathematics 12 (2018), no. 1, 153--165; available online at https://doi.org/10.2298/AADM170405004Q

[4] Feng Qi and Bai-Ni Guo, Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials, Mediterranean Journal of Mathematics 14 (2017), no. 3, Article 140, 14 pages; available online at https://doi.org/10.1007/s00009-017-0939-1

[5] Feng Qi, Da-Wei Niu, Dongkyu Lim, and Yong-Hong Yao, Special values of the Bell polynomials of the second kind for some sequences and functions, Journal of Mathematical Analysis and Applications 491 (2020), no. 2, Paper No. 124382, 31 pages; available online at https://doi.org/10.1016/j.jmaa.2020.124382