Talk:Eulerian numbers, triangle of

The enumeration of Eulerian numbers.

I very strongly advise against the enumeration of the Eulerian numbers as given on the page. Everyone working with Eulerian numbers will observe that the enumeration of these numbers has not been standardized. Donald Knuth himself changed his enumeration in the course of time. So extreme care is advisable when reading the literature.

Also Wikipedia does not get things straight in their article, some statements refer to the enumeration given here and some other statements to another enumeration. The situation in the OEIS comments on the sequence is also far from being consistent.

However things changed. In the recent years the convention seems to stabilize (for good reasons). I advise to follow the enumeration below:

n\k	0	1	2	3	4	5	6
0	1
1	1
2	1	1
3	1	4	1
4	1	11	11	1
5	1	26	66	26	1
6	1	57	302	302	57	1
7	1	120	1191	2416	1191	120	1

This is the enumeration of Graham, Knuth and Patashnik, Concrete Mathematics and it is the enumeration of the Digital Library of Mathematical Functions by the American National Institute of Standards and Technology (which is the successor of the famous Handbook) -- compare Table 26.14.1 --. — Peter Luschny 16:48, 23 July 2010 (UTC)

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Thanks for that recommendation!

I felt too that k should start at 0 (the recurrence formula begs for it,) I'll gladly make the changes! I didn't dare change the convention of Euler's Number Triangle on MathWorld (it chose $\scriptstyle A_{n,1}=A_{n,n}=1\,$).

From the recursion rule, with $\scriptstyle k\in \{1,...,d\}\,$:

$E(1,1)=1,\,$

$E(d,k)=[d-(k-1)]\ E(d-1,k-1)+k\ E(d-1,k),\ d\geq 2,\ k\in \{1,...,d\}\,$

so obviously begs for (although less so with the k+1 on the right...) $\scriptstyle k\in \{0,...,d-1\}\,$

$E(0,0)=E(1,0)=1,\,$

$E(d,k)=(d-k)\ E(d-1,k-1)+(k+1)\ E(d-1,k),\ d\geq 2,k\in \{0,...,d-1\}\,$

I'll also use the number triangles standard (n,k) instead of (d,k) where d refered to the dimension in n^d (for which the Eulerian polynomials where a core part of the generating function.)

And the extratriangular 0! = 1 for degree zero too!

I'm updating everything right now!

Thanks

— Daniel Forgues, July 23, 2010

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Also, concerning the Eulerian polynomials, with $\scriptstyle k\in \{1,...,d\}\,$:

$E_{0}(x)=1,\,$

$E_{d}(x)=\sum _{k=1}^{d}E(d,k)\ x^{k-1},\ d\geq 1\,$

is the d ^th Eulerian polynomial [1] whose Eulerian numbers:

$E(d,k)|_{k=1}^{d}=\{1,(2^{d}-d-1),...,(2^{d}-d-1),1\},\ d\geq 1\,$

can be recursively generated with the triangle of Eulerian numbers.

very much begs for $\scriptstyle k\in \{0,...,d-1\}\,$:

$E_{0}(x)=1,\,$

$E_{d}(x)=\sum _{k=0}^{d-1}E(d,k)\ x^{k},\ d\geq 1\,$

is the d ^th Eulerian polynomial [2] whose Eulerian numbers:

$E(d,k)|_{k=0}^{d-1}=\{1,(2^{d}-d-1),...,(2^{d}-d-1),1\},\ d\geq 1\,$

can be recursively generated with the triangle of Eulerian numbers.

Thanks

— Daniel Forgues, July 23, 2010

More info about Eulerian triangle

I've done some exploration on the Eulerian triangle; some of the observations/properties I would like to include here.

I've very limited resource in the moment; but if you are interested you may look at Eulerian Matrix so far. It includes the important explicite (nonrecursice) definition of the A[,]-entries in terms of binomials, a generalization to negative or general row-indices (providing the analogue row-sum-properties - exhibiting a relation to the gamma and to the zeta-function) which may be included in the article without effort, if this is of interest.

Also the Eulerian matrix can be used for a powerful divergent summation procedure - but that's likely not in the scope of the main-article.

Finally I've collected a list of interesting resources in the reference-list.

Gottfried Helms — Gottfried Helms 06:02, 18 November 2010 (UTC)

Connection to XOR

(stems from a draft sequence submittal by Jean-Jacques Vaudroz Dec 2023). The sequence terms were if(prime(1+ prime(n) XOR prime(n-1)), n>1.
Treating the record values of the resulting terms (2, 7, 3, 13, 7, 29, 3, 5, 11, 3, 59, ...) one sees:
2, 7, 13, 29, 59, 127, 251, 1013, (1021), 8191, 16363, 65519, (65521), 131063, 524287, 1048559, 2097143, 4194301,....
Compare these to A000295 (a few values are skipped):
1, 4, 11, 26, 57, 120, 247, 1013,.............8178, 16369, 65519,................131054, 524268, 1048555, 2097130, 4194281,....
The values clearly correlate. Bill McEachen (talk) 00:24, 22 December 2023 (EST)