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User:Peter Luschny
From OeisWiki
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.
 (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
My BLOG on OEIS
Blog means that I try to explain the background of some sequences which I submitted to OEIS.
I republished the article on Eulerian polynomials as a MathML document. You can find this version and some hints for creating webpages with mathematical content on this page.
Sequential musings
The formula is the answer. Now what was the sequence?  
ζ(k − n,1) − ζ(k − n,k + 1)  
Most coherent themes for which I have contributed sequences
Some other themes I took interest in
Topic  Links to my Homepage  Sequences 
Bernoulliirregular and Eulerirregular 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


More about these formulas can be found on my homepage and on my blog.
On the infrastructure of the binomial triangle
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11 
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 Stirling1 triangles  Generalized Stirling2 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(n1)); T2(mul,a,1,7);
Here some findings:
f(n)  op  alpha  T0  T1  T2  T3 
n^{2}  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 OEISwiki
To use MathJax on the OEISwiki proceed as follows: (1) Install the StiX fonts on your system. http://www.stixfonts.org/ (2) Install the browser addon 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 OEISwiki 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=TeXAMSMML_HTMLorMMLfull"; 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.