General Eulerian Numbers and Eulerian Polynomials

Table 3

First values of general Eulerian numbers 𝐴_𝑛,𝑘(𝑎,𝑑).

	k =
	−1	0	1	2	3
n = 0	1
1	𝑑−𝑎	𝑎
2	(𝑑−𝑎)^2	−2𝑎^2+2𝑎𝑑+𝑑^2	𝑎^2
3	(𝑑−𝑎)^3	3𝑎^3−6𝑎^2𝑑+4𝑑^3	−3𝑎^3+3𝑎^2𝑑+3𝑎𝑑^2+𝑑^3	𝑎^3
4	(𝑑−𝑎)^4	−4𝑎^4+12𝑎^3𝑑−6𝑎^2𝑑^2−12𝑎𝑑^3+11𝑑^4	6𝑎^4−12𝑎^3𝑑−6𝑎^2𝑑^2+12𝑎𝑑^3+11𝑑^4	−4𝑎^4+4𝑎^3𝑑+6𝑎^2𝑑^2 +4𝑎𝑑^3+𝑑^4	𝑎^4