Check the attachment... It uses equations that can be found on the WIKIPEDIA page on Bell polynomials and the generalized chain rule formula (Faà di Bruno equation)...
You could replace g(x) with your equation that uses s(n,k) and f(x) becomes f(g(x))=g(x)-m... But I am not sure how g(x) can be derivated at a glance...
If it actually works out algebraically nicely, the equation would give you a correspondance between higher order derivatives and combinatorics.
This formula has been derived, used, and applied in the following papers and preprints:
(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 http://hdl.handle.net/10515/sy5wh2dx6 and http://dx.doi.org/10515/sy5wh2dx6.
(2) Feng Qi and Jiao-Lian Zhao, Some properties of the Bernoulli numbers of the second kind and their generating function, Journal of Difference Equations and Applications (2017), in press. ResearchGate Working Paper (2017), available online at http://dx.doi.org/10.13140/RG.2.2.13058.27848.
(3) Feng Qi and Bai-Ni Guo, Some properties of a solution to a family of inhomogeneous linear ordinary differential equations, Preprints 2016, 2016110146, 11 pages; Available online at http://dx.doi.org/10.20944/preprints201611.0146.v1.
(4) 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.
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
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