Eulerian number

  • For other numbers and sequences, which are named after Leonhard Euler, Euler numbers, see (disambiguation).
  • For the Euler triangle in the spherical geometry see ball triangle.

Named after Leonhard Euler Euler number An, k in combinatorics, also written as E (n, k) or, is the number of permutations (arrangements ) of 1, ..., n in which exactly k elements larger than the previous are, ie containing exactly k ascents. Equivalent to the definition of " small " instead of " greater " and " downs " instead of " rises ". According to another definition of the Euler number of a (n, k) is the number of permutations of 1, ..., n with exactly k maximum monotonically increasing sections, whereby the second parameter is compared with the definition used here is shifted by one: a (n, k) = A k -1.

Euler Triangle

Like the binomial coefficients of Pascal's triangle, the Euler numbers are arranged in the Euler triangle (first line n = 1, first column k = 0; A008292 in OEIS sequence ):

1                            1 1                         1 4 1                      1 11 11 1                   1 26 66 26 1                1 57 302 302 57 1             1 120 1191 2416 1191 120 1          1 247 4293 1561 9 15619 4293 247 1       1 502 14608 88234 156190 88234 14608 502 1    1 ........................ 1 It can be calculated by the following recursion formula for each entry from the two above it:

For n> 0 with A0, 0 = 1 and A0, k = 0 for k ≠ 0 The convention A0, -1 = 1 and A0, k = 0 for k ≠ -1 would be useful, it is in the alternative definition usual.

Properties

Directly from the definition to follow, 0 = 1 and An, n -1 k = An, k for n> 0 and

The Euler numbers with the formula from the binomial coefficients

Are calculated for n, k ≥ 0, in particular

  • Sequence A000295 in OEIS,
  • Sequence A000460 in OEIS and
  • Sequence A000498 in OEIS.

It is the Worpitzky identity ( Worpitzky 1883)

N ≥ 0, where x is a variable and is a generalized binomial coefficient.

A generating function for An, k is

A relationship with the Bernoulli numbers is βm by the alternating sum

Made for m> 0.

Euler polynomials

The Euler- polynomial An ( x ) is defined by

So

From the corresponding equations for the Euler numbers we obtain the recursion formula

And the generating function

Ronald Graham Oren Patashnik Wolfram Research
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