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# User:Peter Luschny

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## Personal

• Some of my mathematical interests can be found on my homepage.
• My favourite sequence is A057641.
• My favourite integer is 32760. Why? See here.
• My favourite sentence is: "Integers are the decategorification of finite sets."
• My favourite fun formula (which I found in 2010) relates Euler's small gamma A001620 with Euler's big Gamma and computes another constant which is also in OEIS.
 ${\displaystyle \left({{\frac {\Gamma (\gamma )}{\Gamma (2\gamma )\Gamma (1/2-\gamma )}}+{\frac {2\Gamma (1-2\gamma )}{\Gamma (1-\gamma )\Gamma (1/2-\gamma )}}}\right)^{2}\left({\frac {\Gamma (\gamma )\Gamma (1/2-\gamma /2)}{\Gamma (\gamma /2)}}\right)^{4}=\pi }$
• (NEW in 2011, now discontinued.) You can follow me on .
• In my contributions I follow Schmidhuber's Beauty Postulate: "Among several patterns classified as "comparable" by some subjective observer, the subjectively most beautiful is the one with the simplest (shortest) description, given the observer's particular method for encoding and memorizing it." "For example, mathematicians find beauty in a simple proof with a short description in the formal language they are using."
• If you have any comments or suggestions, you can post them on my user talk page.

# Topics on OEIS

## My BLOG on OEIS

Blog means that I try to explain the background of some sequences which I submitted to OEIS.

 Blog pdf new sequences July 2016 Orbitals April 2016 Extensions of the Binomial February 2016 The P-Transformation May 2015 The Bell Transformation December 2014 GFUN - A Sage Toolbox for the OEIS, part 1 July 2014 Binary Quadratic Forms. July 2014 The Unofficial Guide to Coding for the OEIS July 2013 The Bernoulli Manifesto June 2013 Generalized Bernoulli Numbers and Polynomials. May 2013 The Stirling-Frobenius numbers. April 2013 Eulerian polynomials generalized. October 2012 Odds and Ends. October 2012 Aigner triangles. June 2012 The Catalan-Seidel connection. May 2012 The computation and asymptotics of the Bernoulli numbers. April 2012 An old operation on sequences: the Seidel transform. March 2012 Sage in the context of OEIS: tools, tips and tricks. February 2012 The von Mangoldt Transformation January 2012 Transformations of Integer Sequences December 2011 Fibonacci Meanders November 2011 Meanders and walks on the circle October 2011 Perfect and optimal rulers August 2011 The family of Euler-Bernoulli numbers July 2011 Multifactorials June 2011 The lost Pascal numbers May 2011 The lost Catalan numbers April 2011 Set partitions March 2011 Optimal Rulers March 2011 Approximations to the Factorial Function A182914 | A182915 A182919 | A182920 February 2011 Schinzel-Sierpinski conjecture and the Calkin-Wilf tree January 2011 Integer partition trees A182937 December 2010 Double enumerations A000000 | A000000 December 2010 Riemann Hypothesis and the Lagarias Formula December 2010 Extended tables of A094348 and A181852 A094348 November 2010 Strong coprimality A000000 | A000000 November 2010 Swinging Primes A000000 | A000000 November 2010 Prime factors of the swinging factorial A000000 | A000000 October 2010 Notation matters August 2010 Generalized Binomial Coefficients A180274 August 2010 Eulerian Number, a style study A180056 | A180057 August 2010 Eulerian polynomials, initial setup of an article on the wiki, see also the notes below. [x,] A179929 | A179995 August 2010 Figurate number - a very short introduction A179927 | A179928 July 2010 Permutation trees [x,] A179454 | A179455 A179456 | A179457 July 2010 Permutation types [x,] June 2010 Euler's Totient Function A179179 May 2010 Swiss-Knife Polynomials and Euler Numbers [x,] A177762 April 2010 Zeta Polynomials and Harmonic Numbers [x,] A000000 | A000000A000000 | A000000 April 2010 Bernoulli and Worpitzky numbers [x,] A176276 | A176277 Mar 2010 Stern's diatomic array and Binary Partitions [x,] A174980 | A174981 Feb 2010 How I found a Guy Steele sequence [x,] A000000 | A000000A000000 | A000000

I republished the article on Eulerian polynomials as a MathML document. You can find this version and some hints for creating web-pages with mathematical content on this page.

## Most coherent themes for which I have contributed sequences

 Topic Links to my Homepage Sequences Perfect and optimal rulers Generating and counting perfect rulers. A103299 A103301 A103300 A103295 A103296 A103294 A103297 A103298 A004137 The Swiss-Knife polynomials Related to the family of Euler-Bernoulli polynomials. A153641 A162660 A109449 A154341 A154342 A154343 A154344 A154345 Von Staudt prime numbers and generalized Clausen numbers Von Staudt primes, generalized Clausen A092307 A152951 A152952 A141056 A090801 A160035 A160014 Counting with Partitions Partitions of an integer and generalized Stirling triangles of the first and the second kind. A157386 A157385 A157384 A157383 A157400 A157391 A157392 A157393 A157394 A157395 A157396 A157397 A157398 A157399 A157400 A157401 A157402 A157403 A157404 A157405 Variants of Variations A Generalization of the Factorial Function related to Variations and Arrangements, a new scheme. A128195 A128196 A128198 A126062 A126063 A126064 A094587 A007526 A000522 A010842 A090802 A001147 A005917 A014480 A097801 A067311 Swinging factorial Swinging factorial function. A056040 A163865 A163840 A163843 A163650 A163770 A163773 A163649 A163590 A163641 A163644 A163841 A163844 A163771 A163774 A163869 A163842 A163845 A163872 A163772 A163775 A163945 A163640 A163085 A163086 Swinging primes Swinging primes. A163074 A163075 A163076 A163077 A163078 A163079 A163080 A163081 A163082 A163083 A163210 A163211 A163212 A163213 A163209

## Some other themes I took interest in

 Topic Links to my Homepage Sequences Bernoulli-irregular and Euler-irregular primes The computation of irregular primes. A000928 A120337 A128197 Gamma Function On Stieltjes' Continued Fraction for the Gamma Function. A005146 A005147 Lcm{1,2,...,n} The least common multiple as a product of values of the sine function sampled over half of the points in a Farey sequence. See also this paper. A003418 Zumkeller Numbers Interesting partitions of the set of divisors of n. A171641 A171642

## Stirling's famous formula and company

Approximation formulas to the Gamma function.
 Formula Numerator Denominator Stirling A001163 A001164 De Moivre A182935 A144618 NemesG A182912 A182913 Wehmeier A182916 A182917 --New-- A182914 A182915
 Formula Numerator Denominator Stieltjes A005146 A005147 Lanczos A090674 A090675 Nemes A181855 A181856 Gosper A182919 A182920

More about these formulas can be found on my homepage and on my blog.

## On the infrastructure of the binomial triangle

 1 ${\displaystyle \rightarrow \!}$ 2 ${\displaystyle \nearrow \searrow \!}$ 3 ${\displaystyle \nearrow \searrow \nearrow \searrow \!}$ 4 ${\displaystyle \nearrow \rightarrow \rightarrow \searrow \!}$ 5 ${\displaystyle \nearrow \searrow \nearrow \searrow \nearrow \searrow \!}$ 6 ${\displaystyle \nearrow \searrow \rightarrow \rightarrow \nearrow \searrow \!}$ 7 ${\displaystyle \nearrow \searrow \nearrow \searrow \nearrow \searrow \nearrow \searrow \!}$ 8 ${\displaystyle \nearrow \searrow \rightarrow \rightarrow \rightarrow \rightarrow \nearrow \searrow \!}$ 9 ${\displaystyle \nearrow \searrow \nearrow \rightarrow \rightarrow \rightarrow \rightarrow \searrow \nearrow \searrow \!}$ 10 ${\displaystyle \nearrow \searrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \rightarrow \nearrow \searrow \!}$ 11 ${\displaystyle \nearrow \searrow \nearrow \searrow \nearrow \searrow \nearrow \searrow \nearrow \searrow \nearrow \searrow \!}$

This is a symbolic representation of an infrastructure of the binomial triangle characterizing primes. A formal description of the idea can be found in A182929.

## An index of generalized Stirling triangles

Two convenient indices of generalized Stirling triangles. I intend to add cross references on the sequence pages.

 I N D E X Generalized Stirling-1 triangles Generalized Stirling-2 triangles

## Tagging Matryoshka sequences

### Elementary cases

 Type Sequence generator T0 seq(i,i=alpha..beta); T1 seq(op(i,i=alpha..k),k=alpha..beta); T2 seq(op(op(i,i=alpha..k),k=alpha..n),n=alpha..beta); T3 seq(op(op(op(i,i=alpha..k),k=alpha..n),n=alpha..m),m=alpha..beta);

seq means sequence. alpha is typically 0 or 1. beta is a convenient number greater alpha. op is an operation to be specified.

 op alpha T0 T1 T2 T3 add 0 A001477 A000217 A000292 A000332 multiply 1 A001477 A000142 A000178 A055462 sequence 0 A001477 A002262 A056558 A127324 sequence 1 A000027 A002260 sequence -1 A023443 A114219

### Matryoshkas of functions

The elementary cases above are Matryoshka sequences related to the identity function. In the general case the innermost sequence seq(i,i=a..b) becomes seq(f(i),i=a..b). This relates sequences in a new and sometimes surprising way.

With Maple:

T0 := proc(op,f,a,b) local i;
seq(f(i),i=a..b) end;
T1 := proc(op,f,a,b) local k;
seq(op(f(i),i=a..k),k=a..b) end;
T2 := proc(op,f,a,b) local n;
seq(op(op(f(i),i=a..k),k=a..n),n=a..b) end;
T3 := proc(op,f,a,b) local m;
seq(op(op(op(f(i),i=a..k),k=a..n),n=a..m),m=a..b) end;


Example calls:

T3(mul,n->n,1,7);
T2(seq,n->n,0,4);
T1(add,n->n!,0,7);
T1(mul,n->n!,1,7);
a := n -> if(n=0,1,n^(n mod 2)*(4/n)^(n+1 mod 2)*a(n-1));
T2(mul,a,1,7);


Here some findings:

 f(n) op alpha T0 T1 T2 T3 n2 add 0 A000290 A000330 A002415 A005585 multiply 1 A001044 A055209 sequence 0 A133819 sequence 1 A143844

 f(n) op alpha T0 T1 T2 T3 n! add 0 A000142 A003422 A014144 A152689 multiply 1 A000178 A055462 A057527

## Using MathJax on the OEIS-wiki

To use MathJax on the OEIS-wiki proceed as follows:

(1) Install the StiX fonts on your system.
http://www.stixfonts.org/

(2) Install the browser add-on Greasemonkey.
For example for Firefox it is here:
https://addons.mozilla.org/de/firefox/addon/greasemonkey/

(3) Put the user script below in a file named mathjax.user.js.
Install the script. More information here:
http://wiki.greasespot.net/Greasemonkey_Manual:Installing_Scripts

(4) Test
http://oeis.org/wiki/Eulerian_polynomials

(5) Adapt if necessary: Place the mouse on a formula,
open the context menu with a right click and set
(for instance) Settings/ScaleAllMath to 112.

// ==UserScript==
// @name           MathJax on OEIS
// @namespace      http://www.mathjax.org/
// @description    Insert MathJax into OEIS-wiki pages
// @include        http://oeis.org/wiki/*
// ==/UserScript==

if ((window.unsafeWindow == null ? window : unsafeWindow).MathJax == null) {
//
//  Replace the images with MathJax scripts of type math/tex
//
var images = document.getElementsByTagName('img'), count = 0;
for (var i = images.length - 1; i >= 0; i--) {
var img = images[i];
if (img.className === "tex") {
var script = document.createElement("script"); script.type = "math/tex";
if (window.opera) {script.innerHTML = img.alt} else {script.text = img.alt}
img.parentNode.replaceChild(script,img); count++;
}
}
if (count) {
//
//  Load MathJax and have it process the page
//
var script = document.createElement("script");
script.src = "http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML-full";
var config = 'MathJax.Hub.Startup.onload()';
if (window.opera) {script.innerHTML = config} else {script.text = config}
document.getElementsByTagName("head")[0].appendChild(script);
}
}


• Stock includes new unpublished sequences.