I encountered the following triangle of positive integers:

|$c_{n,k}$ | $n=1$ | $n=2$ | $n=3$ | $n=4$ | $n=5$ | $n=6$ | $n=7$ | $n=8$|
|:------:|:------:|:------:|:------:|:------:|:------:|:------:|:------:|:------:|
|$k=0$ | $1$ | $3$ | $15$ | $105$ | $315$ | $3465$ | $45045$ | $45045$|
|$k=1$ || $5$ | $40$ | $385$ | $1470$ | $19635$ | $300300$ | $345345$|
|$k=2$ ||| $33$ | $511$ | $2688$ | $45738$ | $849849$ | $1150149$|
|$k=3$ |||| $279$ | $2370$ | $55638$ | $1317888$ | $2167737$|
|$k=4$ ||||| $965$ | $36685$ | $1200199$ | $2518087$|
|$k=5$ ||||| | $11895$ | $631540$ | $1831739$|
|$k=6$ ||||| | | $169995$ | $801535$|
|$k=7$ ||||| | | | $184331$|

What is the generating function of the sequence $c_{n,k}$ for $0\le k\le n-1$ and $n\in\mathbb{N}=\{1,2,3,\dotsc\}$? Can one find an explicit expression of the sequence $c_{n,k}$ for $0\le k\le n-1$ and $n\in\mathbb{N}$?