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
The line below marks the end of the <noinclude> ... </noinclude> section.
The triangle of
Eulerian numbers (for concatenated rows
{1, 1, 1, 1, 4, 1, 1, 11, 11, 1, ...}
, see
A008292)
-
|
|
|
|
|
|
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| 1
|
|
1
|
|
1
|
| 2
|
|
1
|
1
|
|
2
|
| 3
|
|
1
|
4
|
1
|
|
6
|
| 4
|
|
1
|
11
|
11
|
1
|
|
24
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| 5
|
|
1
|
26
|
66
|
26
|
1
|
|
120
|
| 6
|
|
1
|
57
|
302
|
302
|
57
|
1
|
|
720
|
| 7
|
|
1
|
120
|
1191
|
2416
|
1191
|
120
|
1
|
|
5040
|
|
|
|
|
2
|
3
|
4
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5
|
6
|
7
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|
|
is given by the coefficients of the Eulerian polynomials
-
| En(x) = E (n, m) x n − m, n ≥ 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 is the number of
permutations of the numbers
1 to
in which exactly
elements are greater than the previous element.
The subsequence of Eulerian numbers greater than
1, which are those not lying on the border of the triangle,
i.e. with
, is
{4, 11, 11, 26, 66, 26, ...}
(
A014449).
Example
For
, the sequence
= {1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, ...} = A000583 has the
generating function
-
G{k 4, k ≥ 1}(x) = = | x (x 3 + 11 x 2 + 11 x + 1) | | 1 − 5 x + 10 x 2 − 10 x 3 + 5 x 4 − x 5 |
. |
