Could you please tell me what and where the inverse of a concrete lower triangular matrix showed in the picture is? Can you help compute the inverse matrix explicitly?
In my opinion, my expression is correct. For example, $m_{1,2}=\binom{2X1-2-1}{1-1}=\binom{-1}{0}=0$, $m_{1,3}=\binom{2X1-3-1}{1-1}=\binom{-2}{0}=0$, and $m_{1,4}=\binom{2X1-4-1}{1-1}=\binom{-3}{0}=0$. Is it okay?
Just now I found that the answer to this question existed in my previous preprints below. I tried to delete this question, but failed. Sorry for my neglect.
F. Qi, The inverse of a matrix and several identities related to the Catalan numbers and the Chebyshev polynomials of the second kind, ResearchGate Presentation (2017), available online at https://doi.org/10.13140/RG.2.2.22832.46081/1 (Chinese)
F. Qi, Q. Zou, and B.-N. Guo, Some identities and a matrix inverse related to the Chebyshev polynomials of the second kind and the Catalan numbers, Preprints 2017, 2017030209, 25 pages; Available online at https://doi.org/10.20944/preprints201703.0209.v2
I have drafted a general result for the inverse of the above lower triangular integer unit matrix. After a while, I will anounce my concrete conclusion.
Please read the following preprint for the answer to this question:
Feng Qi, Xiao-Long Qin, and Yong-Hong Yao, The generating function of the Catalan numbers and lower triangular integer matrices, Preprints 2017, 2017110120, 12 pages; Available online at https://doi.org/10.20944/preprints201711.0120.v1 or http://www.preprints.org/manuscript/201711.0120/v1
Please download by yourself the full version from the following address
Feng Qi, Xiao-Long Qin, and Yong-Hong Yao, The generating function of the Catalan numbers and lower triangular integer matrices, Preprints 2017, 2017110120, 12 pages; Available online at https://doi.org/10.20944/preprints201711.0120.v1