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User:Peter Luschny

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  • 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.
\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 sequitter.
  • 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


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

  Blog pdf new sequences
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

Motzkin and Riordan Numbers

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 | A000000
A000000 | 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 | A000000
A000000 | 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.

Sequential musings

The formula is the answer. Now what was the sequence?
 \binom{n}{k}    \left[{n\atop {n+1-k}}\right]   k + \binom{k}{2}(n-k)
  \binom{n}{k}(-2)^{n-k}E_{n-k}(1/2)   \binom{n}{k} (-2)^{n-k} E_{n-k}(1)   \binom{n}{k}2^{n-k}(E_{n-k}(1/2)+E_{n-k}(1))
  \sum_{j=0}^{k}\frac{(-1)^{k-j}}{k+1}\binom{k+1}{j+1}(j+1)^{n+1}   \frac{2k+1}{n+k+1}\binom{2n}{n-k}   \frac{1}{k+1} \binom{n}{k}\binom{n-1}{k}
 \sum_{j=0}^k (-1)^j\binom{n+1}{j}(2(k-j)+1)^n   \prod_{(p+k) \mid (n-k) } p \quad (p\ {prime})  ζ(kn,1) − ζ(kn,k + 1)
 \sum_{j=0}^k(-1)^j\binom{n+1}{j}(k-j+1)^n   {d_n =\text{denom}(B_{n+1}(z+1)-B_{n+1}(1))} \atop {[x^k] \ d_n \sum_{j=0}^{n}\binom{n+1}{j+1}B_{n-j}(1)(x-1)^j}
 {T(0,0) \leftarrow 1; \ T(n,n)\leftarrow T(n-1,0)} \atop {T(n,k)\leftarrow T(n,k+1)+T(n-1,k)}   {T\leftarrow 1; \ m\leftarrow n-k} \atop {\circlearrowleft_{1\leq l\leq k} T \leftarrow (ml+1)(m^l+T)}

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 A001163A001164
De Moivre A182935A144618
NemesG A182912A182913
Wehmeier A182916A182917
--New-- A182914A182915
Formula Numerator Denominator
Stieltjes A005146A005147
Lanczos A090674A090675
Nemes A181855A181856
Gosper A182919A182920

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

On the infrastructure of the binomial triangle

1\rightarrow \!
2\nearrow\searrow \!
3\nearrow\searrow\nearrow\searrow \!
4\nearrow \rightarrow  \rightarrow \searrow \!
5\nearrow\searrow\nearrow\searrow\nearrow\searrow \!
6\nearrow\searrow \rightarrow \rightarrow \nearrow\searrow \!
7\nearrow\searrow\nearrow\searrow\nearrow\searrow\nearrow\searrow \!
8\nearrow\searrow \rightarrow \rightarrow \rightarrow \rightarrow \nearrow\searrow \!
9\nearrow\searrow\nearrow \rightarrow \rightarrow \rightarrow \rightarrow \searrow\nearrow\searrow \!
10\nearrow\searrow \rightarrow\rightarrow\rightarrow\rightarrow\rightarrow\rightarrow\nearrow\searrow\!
11\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.

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:

a := n -> `if`(n=0,1,n^(n mod 2)*(4/n)^(n+1 mod 2)*a(n-1)); 

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.

(2) Install the browser add-on Greasemonkey. 
For example for Firefox it is here:

(3) Put the user script below in a file named mathjax.user.js.
Install the script. More information here:

(4) Test

(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
// @description    Insert MathJax into OEIS-wiki pages
// @include*
// ==/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 = "";
    var config = 'MathJax.Hub.Startup.onload()';
    if (window.opera) {script.innerHTML = config} else {script.text = config}

  • Stock includes new unpublished sequences.
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