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.

$\left({\frac{\Gamma(\gamma)}{\Gamma(2\gamma)\Gamma(1/2-\gamma)}+\frac{2\Gamma(1-2\gamma)}{\Gamma(1-\gamma)\Gamma(1/2-\gamma)}}\right)^2 \left(\frac{\Gamma(\gamma)\Gamma(1/2 - \gamma/ 2)}{\Gamma(\gamma /2)}\right)^4 = \pi$

(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

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
May 2015	Marginalia
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	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?
$\binom{n}{k}$	$\left[{n\atop {n+1-k}}\right]$		$k + \binom{k}{2}(n-k)$
$\binom{n}{k}(-2)^{n-k}E_{n-k}(1/2)$	$\binom{n}{k} (-2)^{n-k} E_{n-k}(1)$		$\binom{n}{k}2^{n-k}(E_{n-k}(1/2)+E_{n-k}(1))$
$\sum_{j=0}^{k}\frac{(-1)^{k-j}}{k+1}\binom{k+1}{j+1}(j+1)^{n+1}$	$\frac{2k+1}{n+k+1}\binom{2n}{n-k}$		$\frac{1}{k+1} \binom{n}{k}\binom{n-1}{k}$
$\sum_{j=0}^k (-1)^j\binom{n+1}{j}(2(k-j)+1)^n$	$\prod_{(p+k) \mid (n-k) } p \quad (p\ {prime})$		$ζ(k - n,1) - ζ(k - n, k + 1)$
$\sum_{j=0}^k(-1)^j\binom{n+1}{j}(k-j+1)^n$		${d_n =\text{denom}(B_{n+1}(z+1)-B_{n+1}(1))} \atop {[x^k] \ d_n \sum_{j=0}^{n}\binom{n+1}{j+1}B_{n-j}(1)(x-1)^j}$
${T(0,0) \leftarrow 1; \ T(n,n)\leftarrow T(n-1,0)} \atop {T(n,k)\leftarrow T(n,k+1)+T(n-1,k)}$		${T\leftarrow 1; \ m\leftarrow n-k} \atop {\circlearrowleft_{1\leq l\leq k} T \leftarrow (ml+1)(m^l+T)}$

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	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	$\rightarrow \!$
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.

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.

User:Peter Luschny