DroneBetter's personal website

physically residing in the UK for now

my humansona is Natalia L. Skirrow (i haven't decided what the L. will mean yet) and it is the general name to use for my corporeal form

on the internet, i am ideally an it, but if you have only known me online i won't be upset at any other pronoun use

being an it is unfortunately untenable in real life; i am somewhat averse to they due to the phenomenon of bigots using degendering as a tool for making trans women more palatable, and i have been subjected to too much bad-faith following of experimental they/themhood exclusively

i consider myself spiritually adjacent to the people who use 'this one' instead of first-person pronouns but have not the wherewithal to do so

criteria for inclusion in this list are not impressiveness (they are mostly rote, with the right background) but surprisingness (to someone without the critical piece of background) that they should be possible at all

refining Stirling's approximation with the first term for its \(\log\)'s Euler-Maclaurin Laurent series (turning it from a lower to an upper bound on the factorial) makes it accurate enough that its errors' product converges; this product is closer to 1 than you may expect!

=\[\prod_{n=1}^\infty\frac{\sqrt{\tau n}\left(\frac ne\right)^ne^{\frac1{12n}}}{n!}=\frac{e^{\frac{1+\gamma}{12}-2\zeta'(-1)}}{\sqrt[4]\tau}<1+\frac1{355}\]
as explained in my page

counting connected series-parallel partial orders on labelled sets with Lagrange inversion

(k+1)=\[\left[\frac{x^k}{k!}\right]\left(\frac x{x+2(1-\cosh(x))}\right)^{k+1}=\sum_{i=0}^k\frac1{i!}\sum_{d=-i}^i{2i\choose i+d}d^{i+k}(-1)^{k-d}\]
as derived in my page ; i am interested in a shorter route from one side to the other!

a formula for the number of \(n\)-derangements

(n)=\[\begin{aligned}!n=\sum_{k=0}^{n-1}(-1)^k{n!\over k!}=\left\lceil n!\over e\right\rfloor&=\frac{n!}e+(-1)^nn\int_{t=0}^\infty{1-e^{e^{-t}-1}\over e^{nt}}\,dt\\&\sim\frac{n!}e+(-1)^n\sum_{k=1}^\infty\frac{(-1)^{k-1}\mathrm{Bell}_k}{n^k}\end{aligned}\]where

\(\lceil x(n)\rfloor=(-1)^n\lceil (-1)^nx(n)\rceil=\begin{cases}\lceil x(n)\rceil&n\ \text{even}\\\lfloor x(n)\rfloor&n\ \text{odd}\end{cases}\) (although here it could be more simply \(!n=\left[n!\over e\right]\)); this makes the exact version with the integral (due to Ramanujan) unnecessary, but still fun

the series (which is a formally-valid asymptotic series/Poincaré expansion) comes out of the Laplace transform via

however, it's possible to derive the series (by some clever rearrangements) and obtain the integral by assuming this is a case in which Watson's lemma's converse holds (a surprisingly viable strategy, even when Watson's lemma does not produce a convergent series; one can for instance obtain Binet's formula for Stirling's approximation's error from its series that comes out of Euler-Maclaurin)

a formula for the number of cycles in all \(n\)-derangements

(n)=\[\begin{aligned}\sum_{k=0}^{n-1}(-1)^k{n!\over k!}(H_{n-k}-1)&=\frac{n!}e\left(H_n-1+\int_{t=0}^\infty\frac{1-e^{1-e^t}}{(e^t-1)e^{nt}}-\frac{e^{-t}\gamma(n,-1)-e^{1-e^t-nt}\gamma(n,-e^t)}{(e^t-1)\Gamma(n)}\ dt\right)\\&=\left\lceil\frac{n!}e\left(H_n-1+\int_{t=0}^\infty\frac{1-e^{1-e^t}}{(e^t-1)e^{nt}}\ dt\right)\right\rfloor\\&\sim\frac{n!}e\left(H_n-1+\sum_{k=0}^\infty\frac{\int_{x=-1}^0\mathrm{Bell}_k(x)}{n^{k+1}}\right)\end{aligned}\]as shown in
I derived the third, inverse-Watsoned to the second, which i discovered experimentally was \(\lceil\rfloor\)able to the correct value, and chased some inequalities from the first to obtain it; interestingly, the absolute error annihilated by the \(\lceil\rfloor\) converges monotonically to 1, so it cannot be turned into a round.

# paths \(P\in\{(1,0,0),(1,1,0),(1,0,1),(1,1,1)\}^n\) that remain confined to the \(x\ge y+z\) tetrahedron at all points

(n,i,j)=\[\binom ni\binom nj-\sum_{m=1}^n\sum_{k=1}^m\frac1k{m-1\choose k-1}{m\choose k-1}{n-m\choose i-k}{n-m\choose j-m-1+k}=\begin{cases}\binom ni\binom nj-\binom n{i-1}\binom n{j-1}&n\ge i+j\\0&\text{else}\end{cases}\]i found the left side in this answer then was surprised to learn of the right side (due to Richard K. Guy, 1991)

books i recommend

Herbert S. Wilf, generatingfunctionology. (1990,2004)

George Pólya, Gábor Szegő, Problems and Theorems in Analysis. (1925)

Ronald Graham, Donald Knuth, Oren Patashnik, Concrete Mathematics. (1989)

Doron Zeilberger,

with Marko Petkovsek, Herbert S. Wilf. A=B (1996)

on how to mechanically prove hypergeometric sum identities (and also find D-finite recurrence relations of hypergeometric sums that aren't necessarily made nice by identities, by exploiting closure properties' guarantees of their existence; this is useful for the next entry,)

with Jet Wimp. Resurrecting the Asymptotics of Linear Recurrences (1985)

on an extremely useful fact; if you know a D-finite recurrence for a sequence, you can get asymptotics for that recurrence's "eigensequences," and their refinements with formally-valid Laurent series coefficients, entirely mechanically (though sometimes it takes a while). (Zeilberger has a Maple package for running this, and the RISC have a port to Mathematica.)

see also things to remember which is somewhat broader

copy all of my writings on my OEISwiki account over

the mainspace has been quite neglected, and the new Managing Editor sees it as out-of-scope

I'm not a fan of "mainspace" Wiki and do not see them as a core part of the OEIS. For explanatory information, Wikipedia is a better place. Rather I think of these pages as supporting any curious conventions that might exist in the OEIS and perhaps as a mechanism for linking the main sequence pages. But that's just my personal opinion, not an "official" opinion whatever that might mean. Most of us ignore the Wiki altogether, which means you are going to have a hard time finding someone prepared to review changes. If you want to make changes there, I would suggest trying to find out the current author and interact with them.

One more general warning is that at some point we are likely to drop PmWiki altogether and perhaps replace it with a simple markdown renderer. It's one of the most difficult pieces of our infrastructure to maintain. If that happens an effort will be made to preserve content, but the more exotic the features used the more likely the pages are to break if that transition does occur.

-Sean A. Irvine, email correspondence (2025-09-14)

this manifested when a change made on 2025-10-06 broke quite a lot of rendering (ie. the positions of bounds for integrals in exponents), so I may expediate it

document my personal Python library dronery here

for some modules this is somewhat already done! linRecur is explained by , and perms by

a page on linear algebra (which is mostly in the matrix module) would be beneficial, since there are a lot of cool nontrivial things for Markov matrices (like obtaining waiting time via the matrix fundamental)

HTMLery

figure out how to make <details> tags function as linkable sections by default

make links inside <abbr> tags render with dotted underlines to indicate that they still have hovertext

miaow

add MathJax (as recommended by David Eppstein)

my friend darcy says KaTeX is better so i may perhaps switch to that; however, this is already much better than Wikipedia's and the OEISwiki's implementations (which just render SVGs instead of HTML)

for pages here that include code, use highlight.js for syntax highlighting my pythonry is quite bracketuous so benefits from bracket pair colourisation, my acquaintance dragoncoder047 informed me prism.js supports this so that is what i use

ConwayLife

when conducting preliminary investigations into CA growth rates for (what eventually grew into) my page , i had some revelations upon the "logarithmic" replicator rule's replicator's growth rate (actually sqrtic); i gave a brief history and eventually the one who had mistakenly believed logarithmitude chimed in to say the rule's name should be changed, so the rule's name was changed... splendid.

note that my investigation was the easiest part; AforAmpere found the emulator that compresses the problem in space and time to make it fit in a Python one-liner, and I believe David Eppstein would have realised its growth rate during his investigations in the 1990s if he had that at his disposal too

Mathematica

a few bugs found that it seems no-one ever encountered before! see this page

i appreciate the convention of don't ask whether you can ask something

i am firmly entrenched in asking culture; if i make a request, please be as forward as possible in your reasons for declining it

if you have two reasons, one being "I don't want to" and the other a more palatable excuse, do not serve me the latter and leave the former to subtext; most likely i will miss it and pester you about whether the latter can be fixed/has fixed itself

in general assume i am a robot that is learning what it means to be human and cannot consistently read between the lines, no matter how good it is at spewing witticisms or how much of an air of confidence and suavity that imbues it with

incidentally, this issue has been much more prevalent in irl interactions than on the internet; i have not had enough experience with internet friends turning into irl friends for extended durations to determine whether this is due to selection bias on my part or the fundamentally different natures, i suspect both!

darcy

soulful creature which has taught me a lot about various things

dangerzone: the PDF file format is Turing-complete, and PDFs are able to phone home and in some cases install malware; this scrubs them of that by opening them in a sandboxed virtual machine and scanning them pixel-by-pixel (which seems a somewhat nuclear solution to me)

Blender: useful for more than just 3Dery!

drone btw would you happen to know of a tool that i could input multiple video files to and have overlay them (for Ocmo purposes)
(this is not quite trivial on account of the background parallax, but the foreground is much higher-contrast and sharper so it only needs to be able to track the positions of black regions)

magma blender has video editing and compositing features

drone what if i wanted to hold the environment fixed and have the videos' boxes move around could it do that

magma yes

drone and could it fill in the pieces of the environment not currently in the videos' boxes based on what videos show of it in the past or future

magma most definitely

magma there is not a predefined Button Which Fills In The Pieces Of The Environment Not Currently In The Videos' Boxes Based On What Videos Show Of It In The Past Or Future, but you should be able to use motion tracking information to just place every video frame in the background simultaneously, offset by its motion info, to emulate the same effect

magma by combining some of the compositing/geometry tools

how to safely inject subcutaneously (and related information that is helpful for DIY HRT in the UK specifically)

some notes on the relativity module i'm currently doing

(only first week done thus far (although it was surprisingly dense, covering a lot of special relativity); there are gaps between weeks because for some of them the lecture notes are entirely sufficient and don't require any expoundment; not made as a general resource but perhaps useful as such)