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Notation

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There doesn't seem to be a standard notation, I've seen E(n,k), A(n,k) and angle brackets used in different sources. So for the sake of consistency I'm picking E(n,k) to be used in the article with a mention of the other notations at the beginning of the article.--RDBury (talk) 18:08, 15 May 2009 (UTC)[reply]

After review of some additional sources I'm changing to A(n,k). It's been used in more places than the others and it's more consistent with the An(x) notation for the polynomials.--RDBury (talk) 18:56, 15 May 2009 (UTC)[reply]

source of the Euler-snippet

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Hi I wanted to find the context of the Euler-snippet; wirkstoff, could you please provide the page-number?

Thanks - Gotti 08:04, 31 August 2009 (UTC) —Preceding unsigned comment added by Druseltal2005 (talkcontribs) Found it myself Gotti 20:23, 1 September 2009 (UTC) —Preceding unsigned comment added by Druseltal2005 (talkcontribs) [reply]

page 485f. --84.130.170.107 (talk) 22:21, 6 September 2011 (UTC)[reply]

Identities

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The first of the listed equations in the "Identities"-section was wrong. To be more precise, in the case where n = 0 the left-hand side is a geometric series, yielding as its limit. But the right-hand side of the equation is zero (we have an empty sum there). I adapted the sum on the right-hand side (the elements are now summed to n not n-1). This corrects the mistake for n = 0 and doesn't change anything for n > 0 because for the index m = n the summand involves a factor A(n,n), which is zero and is this zero itself, contributing nothing to the sum. Now, I'm not an expert in combinatorics and never really worked with Euler numbers. I hope someone, who has experience with these can confirm my correction or look it up in the literature. --TheLaeg (talk) 14:21, 29 March 2010 (UTC)[reply]

I adjusted indices of summation in the power-sum identity so it will be valid for n ≥ 0. Zaslav (talk) 20:11, 7 December 2013 (UTC)[reply]

Eulerian numbers vs Eulerian polynomials

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Premise: the definition and the notation for the Eulerian numbers and polynomials is by no means standard. That said, I see no advantage in the choice of the given definition : "[Eulerian numbers of the first kind] are the coefficients of the Eulerian polynomials:

"

An obvious instance of standardization wants that if certain numbers C(n,m), for 0≤m≤n, are called the X-numbers, the corresponding X-polynomial of order n is

Allowing inconsistencies between the notation for the coefficients and the notation for polynomials, and adopting free variations such as or or may be of some little local advantage, but turns out into a complete mess as soon as one treats polynomials of several type at one time (suppose we have to write sum down the sum of three special polynomials An(x)+Bn(x)+Cn(x), each with its own notation for denoting the coefficients, giving for granted that we are able to keep them all by hart). On the same lines, I don't think that it is an optimal choice having the same letter for the coefficients and the polynomials. In this case, they are too similar, with some risk of confusion. I would go for Knuth solution which is also of more practical use. --pma 17:12, 19 November 2010 (UTC)[reply]

More on Identities

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I've added a third identity on the page, along with how to get it. It relates to the picture posted at the introduction of this page (Euler's Eulerian polynomials). I discovered it a couple of weeks ago and was wondering if anyone has seen it before? I have not, but I am not expert on the subject.


1:26 Dec 07 2010 (PST).

OK but you may as well start from

multiply both sides by the coefficient of any entire power series , and sum over ; of course for you can exchange order of summation in the LHS say, provided ), and obtain

but what is the point? You can write down a lot of these identities... --pma 00:44, 8 December 2010 (UTC)[reply]

Oh I see! I guess this one is kind of special because it's the sum of the actual Eulerian polynomials themselves and it relates with the constant e.

5:32am Dec 08 2010. —Preceding unsigned comment added by 98.164.252.55 (talk) 13:33, 8 December 2010 (UTC)[reply]

This article is still lacking of something more relevant: e.g. the recurrence relations for the E. polynomials of first and second kind; it's also lacking of the g.F. of both. So, when this article will be rectified, there is something to add and something to remove! --pma 01:03, 9 December 2010 (UTC)[reply]

That's very strange! I remember earlier this year they did in fact have the recursive definition for them... check in the history, it should be there... [Edit: Hm or maybe not!]

Sorry if I wasted your time, please remove the identity if you feel it is useless.

I think this identity is nice but the proof should be removed. Either it is original research, or it is known and can be cited in the literature. I will place the text below here so it is still available if needed. It might be desirable to remove the identity altogether as I don't see a reason to prefer it to other identities, but I won't make that decision. Zaslav (talk) 21:27, 7 December 2013 (UTC)[reply]

Another interesting identity is given by the following manipulation:

For , the terms on the right side are positive, so we may switch the sum. The terms on the left make a geometric series, and we know that converges. After all of that, we use the above identity to finish the manipulation:

Finally, for we get

Notice that the numerator on the right-hand side is the Eulerian polynomial shown at the top of this page.

Analogy to normal distribution & Pascal's triangle?

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As n increases, do the values of A(n,k)/n! turn out to approximate a well known function arbitrarily well? Thanks, Rich peterson198.189.194.129 (talk) 02:50, 16 March 2012 (UTC)[reply]

Good question, to which I don't know the answer. One observation: you don't have the scaling quite right; we really want to scale vertically by something like (just based on numerics), which is actually very similar to the Pascal case. (If you just divide by n!, everything goes to 0.) --Joel B. Lewis (talk) 02:49, 17 March 2012 (UTC)[reply]
Update: it's normal. See http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2636886#p2636886 --Joel B. Lewis (talk) 19:33, 22 March 2012 (UTC)[reply]
thanks!198.189.194.129 (talk) 18:16, 23 March 2012 (UTC)[reply]

Discrepancy of the tables of Eulerian numbers regarding $A_{0,0}$

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When I use the generating-function for the introduction of the article to create the table of Eulerian numbers, I get that table (indexes (r,c)=(0,0)...(n,m))

[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[1 4 1 0 0 0 0 0 0 0 0 0 0 0 0 0]
[1 11 11 1 0 0 0 0 0 0 0 0 0 0 0 0]

where the element A_{0,0}=1.

When I use the "explicite formula" (from that so-named section) I get

[0 0 0 0 0 0]
[1 0 0 0 0 0]
[1 1 0 0 0 0]
[1 4 1 0 0 0]
[1 11 11 1 0 0]

where A_{0,0}=0. The second version seems to be in use by combinatorists, and even with an prefixed column [1,0,0,0,...], (and called "shifted version", moreover giving it a unit-diagonal), while the first version is also occuring if we use the "polylog"-(or "Li") function rescaled by x and powers of (1-x). Could someone please look at this with the goal to

a) mention it,     
b) make a comment of
         b1) the history of and
         b2) the relevance/rationale in it? 

Note also that the generalization to negative (and even fractional) row-indexes is only possible when the first version is used. — Preceding unsigned comment added by Druseltal2005 (talkcontribs) 07:45, 19 February 2021 (UTC)[reply]

Removal of python code

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I don't see that having python code in the article adds anything particularly useful; it seems to be original research. I will remove it shortly; but below is a copy for historical reference. 67.198.37.16 (talk) 23:55, 5 January 2024 (UTC)[reply]

The following is a Python implementation.
import math  # Python 3.8

def Ank(n, k) -> int:
    """
    Compute A(n, k) using the explicit formula.
    """
    def summand(i):
        return (-1) ** i * math.comb(n + 1, i) * (k + 1 - i) ** n
    return sum(map(summand, range(k + 1)))

def Anks(n) -> list:
    """
    Coefficient list for the n'th polynomial A_n(t).
    """
    return [1] if n == 0 else [Ank(n, k) for k in range(n)]

def eval_polynomial(coeffs, t):
    """
    Polynomial evaluation function.
    """
    return sum(c * t ** k for k, c in enumerate(coeffs))

def An(n, t: float) -> float:
    """
    Polynomial A_n(t).
    """
    return eval_polynomial(Anks(n), t)

# Print the first few polynomials
sup = lambda n: str(n).translate(str.maketrans("0123456789", "⁰¹²³⁴⁵⁶⁷⁸⁹"))
sub = lambda n: str(n).translate(str.maketrans("0123456789", "₀₁₂₃₄₅₆₇₈₉"))

NUM = 8
for n in range(NUM):
    print(f"A{sub(n)}(t) = " + " + ".join(f"{ank} t{sup(k)}" for k, ank in enumerate(Anks(n))))
    # E.g. A₇(t) = 1 t⁰ + 120 t¹ + 1191 t² + 2416 t³ + 1191 t⁴ + 120 t⁵ + 1 t⁶

    # Sanity checks
    assert Ank(n, 1) == 2 ** n - (n + 1)
    assert n == 0 or An(n, 1) == math.factorial(n)  # Cardinality check
    alternating_sum: float = sum((-1)**k * Ank(n, k) / math.comb(n - 1, k) for k in range(n))
    assert n < 2 or abs(alternating_sum) < 1e-13