Intended for: November 4, 2011
Timetable
 First draft entered by M. F. Hasler on November 3, 2011 ✓
 Draft reviewed by Alonso del Arte on April 9, 2012 ✓
 Draft to be approved by October 4, 2012
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A008292: The triangle of Eulerian numbers






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is given by the coefficients of the Eulerian polynomials
 $E_{n}(x)=\sum _{m=1}^{n}E(n,m)\ x^{nm},\quad n\geq 1,\,$
which appear in the numerator of an expression for the
generating function of the sequence
{k n}k ≥ 1 = {1 n, 2 n, 3 n, ...}, n ≥ 1 
.
The
Eulerian number $\textstyle {\left\langle {n \atop m}\right\rangle }$ is the number of
permutations of the numbers
to
in which exactly
elements are greater than the previous element.
The subsequence of Eulerian numbers
, which are those not lying on the border of the triangle,
i.e., with
, is
A014449 .
Example
For
, the sequence
{k n}n = 4, k ≥ 1 = {1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, ...} = 
A000583 has the
generating function
 $G_{\{k^{4},k\geq 1\}}(x)={\frac {x\,E_{4}(x)}{(1x)^{5}}}={\frac {x\,(x^{3}+11x^{2}+11x+1)}{15x+10x^{2}10x^{3}+5x^{4}x^{5}}}~.\,$