Generating function for terms of Euler triangle?

[ktoz]

I'm sure this is relatively easy, but after an hour or so googling, I can't seem to find the formula for generating terms of the http://steiner.math.nthu.edu.tw/chuan/123/test/euler.htm

Is this known by some other name? Maybe that's why I can't find it?

Thanks

 

[ktoz]

Found it at Sloan's

For anyone else who's interested, the formula is:

A(n,k)=Sum (-1)^j*(k-j)^n*C(n+1,j), j=0..k

and the link is here here

 

[tacman]

for your question! The generating function for the terms of the Euler triangle is also known as the Eulerian polynomial. It can be written as:

E(x,t) = 1 + \sum_{n=1}^{\infty} \frac{t^n}{(1-t)(1-2t)...(1-nt)}x^n

where n represents the row number in the Euler triangle and t represents the variable used for the terms in each row. This generating function can be used to find any term in the Euler triangle by plugging in the appropriate values for n and t.

The Euler triangle, also known as the Eulerian triangle or the triangle of Eulerian numbers, is a triangular array of numbers that arise in combinatorics and number theory. Each number in the triangle represents the number of permutations of n objects with k descents (where a descent is defined as a decrease in value between two consecutive objects).

I hope this helps! Keep exploring and learning about the fascinating world of Eulerian numbers and their applications.