Personal

• "Integers are the decategorification of finite sets."
• Some of my mathematical interests can be found on my homepage.
• (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."
• If you have any comments or suggestions, you can post them on my user talk page.
• My favorite 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.

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

Topics on OEIS

• My favorite contribution to the OEIS:

Unfortunately, the editors did not allow the name. However, this list is nothing other than the complement to Georg Cantor's famous list, with which he counted the rational numbers, A352911.

Since we're on the subject of Cantor, there are a number of other enumerations: So the enumeration of the rational numbers in the closed real interval [0, 1], A366191.

Then there is the boustrophedonic Cantor enumeration A319571. If (x, y) and (x', y') are adjacent points on the trajectory of the map then max(|x - x'|, |y - y'|) is always 1 whereas for the Cantor enumeration this quantity can become arbitrarily large. In this sense the boustrophedonic variant is continuous whereas Cantor's original realization is not.

My BLOG on the OEIS

Here I try to explain the background of some sequences which I submitted to OEIS:

Sequential musings

The formula is the answer. Now what was the sequence?
${\displaystyle {\binom {n}{k}}}$		${\displaystyle \left[{n \atop {n+1-k}}\right]}$		${\displaystyle k+{\binom {k}{2}}(n-k)}$
${\displaystyle {\binom {n}{k}}(-2)^{n-k}E_{n-k}(1/2)}$		${\displaystyle {\binom {n}{k}}(-2)^{n-k}E_{n-k}(1)}$		${\displaystyle {\binom {n}{k}}2^{n-k}(E_{n-k}(1/2)+E_{n-k}(1))}$
${\displaystyle \sum _{j=0}^{k}{\frac {(-1)^{k-j}}{k+1}}{\binom {k+1}{j+1}}(j+1)^{n+1}}$		${\displaystyle {\frac {2k+1}{n+k+1}}{\binom {2n}{n-k}}}$		${\displaystyle {\frac {1}{k+1}}{\binom {n}{k}}{\binom {n-1}{k}}}$
${\displaystyle \sum _{j=0}^{k}(-1)^{j}{\binom {n+1}{j}}(2(k-j)+1)^{n}}$		${\displaystyle \prod _{(p+k)\mid (n-k)}p\quad (p\ {prime})}$		${\displaystyle \zeta (k-n,1)-\zeta (k-n,k+1)}$
${\displaystyle \sum _{j=0}^{k}(-1)^{j}{\binom {n+1}{j}}(k-j+1)^{n}}$			${\displaystyle {d_{n}={\text{denom}}(B_{n+1}(z+1)-B_{n+1}(1))} \atop {[x^{k}]\ d_{n}\sum _{j=0}^{n}{\binom {n+1}{j+1}}B_{n-j}(1)(x-1)^{j}}}$
${\displaystyle {T(0,0)\leftarrow 1;\ T(n,n)\leftarrow T(n-1,0)} \atop {T(n,k)\leftarrow T(n,k+1)+T(n-1,k)}}$			${\displaystyle {T\leftarrow 1;\ m\leftarrow n-k} \atop {\circlearrowleft _{1\leq l\leq k}T\leftarrow (ml+1)(m^{l}+T)}}$

Some coherent themes for which I have contributed sequences

Some other themes I took interest in

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

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.

Tagging Matryoshka sequences

Elementary cases

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

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: