I was working on finding the Recurrence Relation of a sequence with a definition Similar to the Central Delannoy Numbers and so i began to study them.
CDN : https://en.wikipedia.org/wiki/Delannoy_number
The CDN obey the following Recurrence Relation :
$$ nD(n) = 3(2n-1)D(n-1)-(n-1)D(n-2)$$
In this post : Combinatorial interpretation of Delannoy numbers formula
I read that the recurrence relation can be obtained using First Step or Last Step Analysis but i don't quite understand the process too well after reading some papers on it.
I also tried to read :
$\textbf{ Peart, Paul; Woan, Wen-Jin (2002). "A bijective proof of the Delannoy recurrence". Congressus Numerantium. 158: 29–33 }$
but was unable to gain access to the full article to be able to look at a proof.
In this paper they do have a proof for a different Recurrence Relation but it is quite difficult for me to follow : https://arxiv.org/pdf/2003.12572.pdf
So my question is : Would somebody please provide the simplest proof you can come up with or know for finding the recurrence relation for the CDN.
Thank you kindly for your help and time.