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Template:Sequence of the Day for November 4

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

 n

 n!

1   1
1
2   1 1
2
3   1 4 1
6
4   1 11 11 1
24
5   1 26 66 26 1
120
6 1 57 302 302 57 1
720
7   1 120 1191 2416 1191 120 1
5040

1
2
3
4
5
6
7

is given by the coefficients of the Eulerian polynomials

${\displaystyle E_{n}(x)=\sum _{m=1}^{n}E(n,m)\ x^{n-m},\quad n\geq 1,\,}$
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
 E (n, m) =
${\displaystyle \textstyle {\left\langle {n \atop m}\right\rangle }}$ is the number of permutations of the numbers
 1
to
 n
in which exactly
 m
elements are greater than the previous element.
The subsequence of Eulerian numbers
 > 1
, which are those not lying on the border of the triangle, i.e., with
 1 < m < n
, is A014449
 = {4, 11, 11, ...}
.

Example

For
 n = 4
, the sequence
 {k n}n  = 4, k    ≥  1 = {1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, ...} =
A000583
 (k), k   ≥   1,
has the generating function
${\displaystyle G_{\{k^{4},k\geq 1\}}(x)={\frac {x\,E_{4}(x)}{(1-x)^{5}}}={\frac {x\,(x^{3}+11x^{2}+11x+1)}{1-5x+10x^{2}-10x^{3}+5x^{4}-x^{5}}}~.\,}$