Talk:Pascal's triangle

Code

Hello, Iprogram with a much shorter, and, I hope, more comprehensible program. The goal here is to help people understand Pascal's triangle. I don't think the gyrations needed to get the numbers to line up were aiding that understanding. I also don't think the standard Java cruft (class name, imports, try/catch) was helping either. Anyway I'm sure that the floodgates are open now, and we're going to have 200 versions of this program in different languages... which is OK by me, as long as everybody is striving towards comprehension of Pascal's triangle. I claim the existing program is an improvement over the previous one. I don't claim it's the best possible. Happy editing, Wile E. Heresiarch 05:57, 8 Jun 2004 (UTC)

As I mentioned on Wikipedia: Wikicode/Specification, I really don't mind you reverting to the Python code; in fact, I probably shouldn't have converted it in the first place, since it's against my policy of leaving real-language code alone. Sorry about that. Derrick Coetzee 17:57, 10 Oct 2004 (UTC)

More about the code. I see the C code has been restored. I don't think this is an improvement. Why don't we just cut out the code altogther (in any language). Having an algorithm to print out some rows of the triangle doesn't have much to do with Pascal's triangle. I originally put in the Python because it replaced a Java program that was about 10 times as long; but in any event having a program is optional, so let's just cut it and avoid the language wars. Wile E. Heresiarch 06:47, 6 Mar 2005 (UTC)

So algorithms (though tied to a specific language) are not encyclopedia worthy? Cburnett 08:36, 6 Mar 2005 (UTC)

I agree with Wile that the code does not add any information on Pascal's triangle. The algorithm, based on Pascal's identity is already explained in English in the lead section, and it is straightforward to translate it in a specific programming language. -- Jitse Niesen 15:23, 6 Mar 2005 (UTC)

I've cut the section with the computer code. As I said in the edit summary, "source code doesn't shed any light on Pascal's triangle, and it's a language war magnet". For what it's worth, Wile E. Heresiarch 15:14, 7 Mar 2005 (UTC)

Question about a listed property

The page lists the following claim:

"the sum of the squares of the elements of the n^th row equals the middle element of the 2n^th."

But it appears to me that the claim should be "the sum of the squares of the elements of the n^th row equals the middle element of the 2n^th-1."

Am I wrong? This assumes that the row numbering starts at one (which is consistent with other parts of the text). In particular, only the odd numbered rows have a "middle element", and odd numbered rows must be of the form 2n-1, not 2n.

I believe you're right — this is an off-by-one error. Deco 08:37, 24 Mar 2005 (UTC)

Some further explanations prompted by the edits of Chad.nezar: The article assumes that the row with just one 1 is row number one. It then follows that row number n has the binomial coefficients

${n-1 \choose 0},{n-1 \choose 1},{n-1 \choose 2},\ldots ,{n-1 \choose n-1}.$

Hence, the text "the sum of the squares of the elements of the n^th row equals the middle element of the (2n - 1)^th" means

$\sum _{k=0}^{n-1}{n-1 \choose k}^{2}={2n-2 \choose n-1}.$

Substituting $m=n-1$ yields

$\sum _{k=0}^{m}{m \choose k}^{2}={2m \choose m},$

which is the formula given in the text. -- Jitse Niesen 10:57, 6 Apr 2005 (UTC)

Jitse, thanks for that example, which helps clarify some things for me. However, it also points out the other inconsistencies of this article. The crux of the problem is that the text uses the letter n to indicate rows as the n^th row, where the row numbering starts at 1. However, clearly the math formulas and notation require that n start at 0. In the example you gave above, you have to define $m=n-1$ for just this purpose.

This truth is demonstrated by the example in this section, showing that squares of the terms of the 5th row (where row numbering starts from 1) add up to 70. However, the math formula demonstrating this sums over k which starts at zero. Thus, if n is equal to 5 (as the text would indicate), that sum has a total of 6 terms, not 5. Clearly it is written assuming that the n^th row numbering starts with the 0^th row.

In the first section, there is also some confusion in that the page states "for positive integers n and k where n ≥ k", however the examples require k to start at 0, and implicitly that n start at zero (otherwise, the case of ${n \choose n}$ is irrelevant). I therefore think the wording should be "for non-negative integers n and k where n ≥ k", with all the math formulas based on n starting at zero, and the text should be updated to not use the phrase "n^th row", but something more appropriate. Either the text should be changed to indicate that we consider the first row to be the 0^th row, or use another variable (like m) to indicate the row number and give it's relationship to n. Or just change n^th row in the text, to $(n+1)$^th row, where needed. Comments? Chad.netzer 22:07, 8 Apr 2005 (UTC)

I think your (Chad.netzer's) edits make the article clearer, so thanks for that. -- Jitse Niesen 11:51, 11 Apr 2005 (UTC)

Pingala

The following material was added by an IP user, and subsequently edited several times; e.g., the spelling "rythem" was corrected to "rythm" (along with other changes), but that was reverted.

Earliest evidence about this triangle was found in the year 200 B.C. Pingalas Sanskrit Text Chandah-Sutra . The triangle was called Meru Prastara. It was used to identify the poetic metre and to combine the short and long syllables to produce the required rythem. Although Biggs did not conclude if this triangle originated by the Hindu scholars, he definitely mentions about further research on the origin of the Arthmatic triangle.'

I have for now removed it; it may be sound (I do not have access to the source), but perhaps we can find an acceptable version here before re-adding it. Nø (talk) 09:59, 21 March 2023 (UTC)

We have several high quality sources in the article that attribute the binomial coefficients and the triangle to medieval Muslim mathematicians and so far, i've never seen similar high quality sources supporting this kind of edits. Besides, some weeks ago, i already removed some similar claims.---Wikaviani ^{(talk) (contribs)} 20:34, 21 March 2023 (UTC)

The cited article is publicly available at https://www.sciencedirect.com/science/article/pii/0315086079900740 , and in fact confirms the material deleted by Nø. So I intend to re-add it and fix the errors. - Jochen Burghardt (talk) 10:12, 22 March 2023 (UTC)

Although "historia mathematica" seems to be in general a reliable source for this kind of topic, I would like to draw your attention to the fact that the author of the article on the other hand is not an expert source on the history of mathematics, so this source cannot be used to counterbalance the other sources in the section which are of much better quality (Rashed, Brummelen, Sidoli are all promminent historians of science and mathematics). Also, Gibbs himself does not claim that the coefficients or the triangle were known to ancient Indian mathematicians, rather, he discusses the matter and states that, given the current state of knowledge, al-Tusi must be recorded as the earliest reference to the triangle. Thus, i see no reason to re-add the removed content.---Wikaviani ^{(talk) (contribs)} 18:13, 22 March 2023 (UTC)

I still didn't read Gibbs' Biggs' article, so I can't yet comment on your last sentences. However, I wonder how you decide who is an expert and who isn't. - Jochen Burghardt (talk) 10:30, 23 March 2023 (UTC)

Biggs apparently has a wikipedia article: Norman L. Biggs. - Jochen Burghardt (talk) 10:40, 23 March 2023 (UTC)

Now I read p.130-131 of Biggs, and I agree with Wikaviani that Biggs claims (al-Tusi 1265) to be the earliest reference. I suggest to try to find the source (Ahmedev and Rosenfeld 1963) given by Biggs, and then add an appropriate remark, quite different from the deleted one. - Jochen Burghardt (talk) 11:43, 23 March 2023 (UTC)

Well, i knew about Biggs' Wiki article, that's where i found that he is a mathematician, and while he has written some papers about the history of maths, he is not a prominent historian of maths simply because that field is not his main field, unlike Roshdi Rashed or Glen Van Brummelen. I have no problem with trying to find the source you mentioned, but honestly, i don't see how the addition of what a single source says about the topic of this article would be a notable improvement of it.---Wikaviani ^{(talk) (contribs)} 22:04, 23 March 2023 (UTC)

most of the article written in science direct is not reliable and many authors claimed to be an expert.I saw a article on cow urine therapy treatment for black fungus in science direct and itself a pseudoscience article. Ppppphgtygd (talk) 21:39, 22 March 2023 (UTC)

A judgment about general reliablilty of a journal can't be based on an (unsourced rumor about a) single article. - Jochen Burghardt (talk) 10:33, 23 March 2023 (UTC)

When I remvoed the fairly recent paragraph on Pingala, I was not aware that other material on pre-islamic insights into binomial coefficients (arranged in a triangle, or not) har been removed in this edit: diff. I wonder if some of that material should be rescued. Nø (talk) 16:14, 6 April 2023 (UTC)

Does this source is acceptable on pingala

http://5010.mathed.usu.edu/Fall2022/EHumes/history.html 122.161.48.206 (talk) 02:06, 2 February 2024 (UTC)

Even if it were, it doesn't support the changes you want to make. It explains that Pascal's Triangle doesn't appear in what survives of Pingala's work, but in Halayudha's more than 1,000 years later. MrOllie (talk) 02:10, 2 February 2024 (UTC)

Actually i am very much interested to know truth even if halyudha discover it his name should definitely come to in history of pascal triangle. Also halyuddha guy give his commentary on pingala work. On indian mathematician wiki pages it was clearly mention pingala discover pascal triangle and halyuddha do commentary on his work

the reference of this content was given by some Fowler and david person.

Fowler, David (1996), "Binomial Coefficient Function", The American Mathematical Monthly, 103 (1): 1–17, doi:10.2307/2975209, JSTOR 2975209

I request you Pls visit indian mathematocs wiki page and see pingala section and revise pascal triangle history accordingly.

Thanne 2409:4050:2D05:5C64:0:0:A89:770F (talk) 03:12, 2 February 2024 (UTC)

The Indian version of Wikipedia is a separate project with vastly different policies and standards. What happens there has no bearing on the English wiki. MrOllie (talk) 03:26, 2 February 2024 (UTC)

Actually it was english wikipedia only thats why i am saying pls check it 2409:4050:2D05:5C64:0:0:A89:770F (talk) 04:30, 2 February 2024 (UTC)

References

OEIS

Should the OEIS sequence for the triangle be linked from this article? E L Yekutiel (talk) 01:13, 3 May 2023 (UTC)

"Meru Prastara" listed at Redirects for discussion

The redirect Meru Prastara has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2023 May 8 § Meru Prastara until a consensus is reached. Pichpich (talk) 22:59, 8 May 2023 (UTC)

This has now been deleted, on the grounds that the phrase "Meru Prastara" no longer is mentioned in the article. This is the result of edits (discussed above) that have completely removed any discussion of Indian mathematics from the history section. Globally it seems clear to me that this is not the right outcome, and that some discussion of the triangle in Indian mathematics should be restored (after which the redirect can be recreated). --JBL (talk) 17:27, 16 May 2023 (UTC)

Addition of a section on Morton's discovery

Hi, I'm considering adding the following section and would like any feedback on whether it is worth mentioning, and if so, whether the description is up to standards. This description would replace the fourth bullet point (The value of a row) under Rows and be appended to the Extensions section. Any pointers to related demonstrations of this would be also be greatly appreciated:

NOTE: Please see my sandbox for the latest revision. — Preceding undated comment added 23:47, 3 June 2023 (UTC)

Arbitrary Bases

File:Pascal-duodecimal.png

Entries of Pascal's Triangle in radix a = 12. Digits at the k-th entry begin to compound when radix a <= n choose k.

In 1964, Dr. Robert L. Morton presented the argument that each row $n$ can be read as a radix $a$ numeral, where $\lim _{n\to \infty }11_{a}^{n}$ is the hypothetical terminal row, or limit, of the triangle, and the rows are its partial products. He proved the entries of row $n$, when interpreted directly as a place-value numeral, correspond to the binomial expansion of $(a+1)^{n}=11_{a}^{n}$. To better understand the principle behind this interpretation, here are some things to recall about binomials:

• A radix $a$ numeral in positional notation (e.g. $14641_{a}$) is a univariate polynomial in the variable $a$, where the degree of the variable of the $i$th term (starting with $i=0$) is $i$. For example, $14641_{a}=1\cdot a^{4}+4\cdot a^{3}+6\cdot a^{2}+4\cdot a^{1}+1\cdot a^{0}$.
• A row corresponds to the binomial expansion of $(a+b)^{n}$. We can eliminate $b$ from the expansion by setting $b=1$. The expansion now typifies the expanded form of a radix $a$ numeral, as demonstrated above. Thus, when the entries of the row are concatenated and read in radix $a$ they form the numerical equivalent of $(a+1)^{n}=11_{a}^{n}$. For example, $14641_{a}$ is $11_{7}^{4}=8^{4}$ in radix $7$, $11_{8}^{4}=9^{4}$ in radix $8$, and so on (in general, if $c=a+1$ for $c<0$, then let $a=\{c-1,-(c+1)\}\;\mathrm {mod} \;2c$, with odd values of $n$ yielding negative row products).

The frequently cited row products $(a+1)^{n}=11_{1}^{n}=2^{n}$ and $(a+1)^{n}=11_{10}^{n}=11^{n}$ are obtained by setting the rows' radix (the variable $a$) equal to one and ten, respectively. As another example, setting $a=n$ yields the row product $n^{n}\left(1+{\frac {1}{n}}\right)^{n}=11_{n}^{n}$. The numeric representation of $11_{n}^{n}$ is formed by concatenating the entries of row $n$. In the image above, the twelfth row denotes the product:

$11_{12}^{12}={\boldsymbol {1}}:{\boldsymbol {1}}0:56:164:353:560:650:560:353:164:56:1{\boldsymbol {0}}:{\boldsymbol {1}}_{12}={\boldsymbol {2}}7433a96997{\boldsymbol {01}}_{12}=2613035290224..._{10}$

with compound digits (delimited by ":") in radix $12$ and a priori values in bold. The digits from $k=n-1$ through $k=1$ are compound because these row entries compute to values greater than or equal to $12$. To normalize the numeral, simply carry the first compound entry's prefix, that is, remove the prefix of the coefficient ${n \choose n-1}$ from its leftmost digit up to, but excluding, its rightmost digit, and use radix-$12$ arithmetic to sum the removed prefix with the entry on its immediate left, then repeat this process, proceeding leftward, until the leftmost entry is reached. In this example, the normalized string ends with $01$ for all $n$. The leftmost digit is $2$ for $n>2$, which is obtained by carrying the $1$ of $10_{n}$ at entry $k=1$. It follows that the length of the normalized value of $11_{n}^{n}$ is equal to the row length, $n+1$. The integral part of $1.1_{n}^{n}$ contains exactly one digit because $n$ (the number of places to the left the decimal has moved) is one less than the row length. Below is the normalized value of $11_{1234}^{1234}$. Compound digits remain in the value because they are radix $1234$ residues represented in radix ten:

$11_{1234}^{1234}=2:885:2:35:977:696:\overbrace {...} ^{\text{1227 digits}}:0:1_{1234}=2717181235..._{10}$

[Removed summary]

Twoxili (talk) 04:09, 30 May 2023 (UTC)

References

Hi Twoxili, as written this has the appearance of being the product of your own original research. Wikipedia is an encyclopedia; the articles here are based on published sources, as described in our policy WP:OR (and other related content policies, like WP:V and WP:DUE). I would be opposed to adding this to our article without evidence that it has been described in reliable published sources. --JBL (talk) 18:15, 30 May 2023 (UTC)

You've now added a reference, but since the cited paper does not mention the number e, it does not improve the situation. --JBL (talk) 20:46, 30 May 2023 (UTC)

Hi JBL. Thanks for providing links to the relevant guidance documents. Yes, I added a source that gives a more thorough treatment to all the points I made except Step 4 (the identity involving the row binomial and the limit expression of e). Otherwise the steps I have listed are a paraphrase of the cited paper. As for the identity in Step 4, it is a trivial observation, meaning, anyone who is farmilar with the cited paper, specifically the attention that is given to the special case of b = 1 in the binomial (a + b)^n, and is also familiar with the limit expression of e, can clearly detect that they are identical, that is, each row is (a + 1) multiplied by itself n many times. I was not able to find much more discussion on the conclusion reached by the cited paper (and I assume this is because it is common knowledge in this domain), but my aim is to present the findings of that paper in a way that helps convey how Morton interprets the rows. My question is, does the cited paper help to resolve your concern? Twoxili (talk) 20:58, 30 May 2023 (UTC)

The paper mentions (a + 1)^n. The limit of this value as n -> ∞ is e scaled to a whole number for a = n. This is what I mean by "trivial observation". Does this help mitigate your concerns? Twoxili (talk) 21:20, 30 May 2023 (UTC)

I've included a summary in the introductory paragraph. I do agree that, based on the Wikipedia guidelines, and given the state of available material, "a = n" is treading into original territory, due to the sparsity of discourse on the topic. As I mentioned, the formula product of which e is a limit (though not mentioned by name) is stated in the published material, and should seem obvious to any student of math if they are familiar with the paper. The cited paper includes a version of Pascal's triangle in base 12 and explicitly states that row n denotes 13^n. It seems Prof. Morton couldn't be bothered to mention the significance of the 12th (or nth $a$th) power of 13 specifically [Edit (not even as a footnote)], nor can I find any discussions about row n of the radix n triangle. As for why that is, my thoughts are that hardly anyone has read the paper (as of today it is cited only 8 times according to Google Scholar), or that "a = n" doesn't add anything to the discourse. To say that the structure of the triangle is e given the definition of the entries as outlined in Morton's paper is to merely restate that definition [Edit: in terms of e]. At any rate, I thank you for your feedback. My understanding of this topic and the issues involved in editing Wikipedia articles are greatly improved by your comments. Twoxili (talk) 12:07, 31 May 2023 (UTC)

To me personally, this seems far from passing the bar set by WP:OR and WP:DUE, but it's possible that other editors will disagree. --JBL (talk) 17:24, 31 May 2023 (UTC)

I made some more edits to remove any original insights or ideas, with the aim of distilling it to the prior result obtained by the cited paper. The discussion of this result is convoluted, that's for sure. The result is alluded to multiple times in the Wikipedia page, but never explicitly asserted. The use of arbitrary bases, and the fact that the subject of the discussion (the row entries) have multiple mathematical objects associated with them (the index, the polynomial term, it's solution, the degree, the coefficient, etc) doesn't help with communicating the mechanism at work, so I focused on clarifying the discussion. As for whether "a = n" is original, at this point I feel it's "prior art" in need of more discussion and clearer framing. [Edit: My only contribution is nomenclature.] Even the Wikipedia page applies the b = 1 case and changes base a to elucidate certain properties, (in the Binomial Theorem section) setting a = 1 to obtain 2^n and (in the Rows section) setting "a = 10" to obtain 11^n. The paper states "The entries of Pascal's triangle can be expressed in any place-value numeral system which we may choose..." to obtain (a + 1)^n = 11^n in the chosen base a. The results 2^n and 11^2 are due to changing the base a. The problem is that the Wikipedia page never mentions this treats the two (the row sum and the base change) as separate phenomena. This property is the thesis of the cited article, and it's something I feel the Wikipedia page is missing. Twoxili (talk) 22:18, 31 May 2023 (UTC)

I've revised the writeup to better highlight the connection between Morton's paper and e. Basically, the sequence 11 (the topic of Morton's paper) is the root of e. This wasn't made clear in the paper, perhaps because it's unremarkable at first glance. In the comments below, in response to Mgnbar's comments, I presents some arguments for why I feel this is important to explicitly capture in the discussion about Pascal's triangle. Twoxili (talk) 17:01, 2 June 2023 (UTC)

Let me see whether I can summarize this argument concisely. The nth row of the triangle is

${n \choose 0},{n \choose 1},\cdots ,{n \choose n}.$

If we interpret those integers as digits in a base-n expansion of another integer — let's call it k — then

$k={n \choose 0}n^{n}+{n \choose 1}n^{n-1}+\cdots +{n \choose n}=(n+1)^{n}=n^{n}\left(1+{\frac {1}{n}}\right)^{n}.$

As $n\to \infty$, the $(1+1/n)^{n}$ on the far right limits to e, while the $n^{n}$ on the far right limits to infinity. So k limits to infinity, and $kn^{-n}$ limits to e.

First question: Am I correctly summarizing the argument?

Second question: If so, then is the result $kn^{-n}\to e$ notable? Mgnbar (talk) 19:00, 1 June 2023 (UTC)

Hi Mgnbar. You are correct in summarizing the argument. The value k is infinitely large (it has no decimal places). The statement $kn^{-n}\to e$ says "the hypothetical bottom row of Pascal's triangle in base $n$ is, on the first order of magnitude, $e$". As to your question of whether this is notable, this is also my question. Here are my thoughts. Firstly, when I search for "e in Pascal's triangle", the ratio of the row products turn up, but that's it. I've never heard anyone mention this explicitly, though I suspect it's probably taken for granted by most. While the result may not be very surprising, I think it deserves to be said for anyone just wanting to know all the ways e is connected to the triangle. What the argument says to me is that, as you calculate the rows from top to bottom, you're computing e. The triangle computes e. The casual student might find that quite interesting. This convergence is what the cited paper was getting at, but it was written without the benefit of the better framework we have today for discussing the triangle. At least for me, seeing the rows as powers of the sequence 11, and knowing that the root of e is the sequence 1.1 gives me a certain satisfaction with regards to wanting to understand what, if any, connections there are between e and the triangle. I'd be very interested to hear your thoughts about this. * This has been edited to clarify the meaning of the original comment. Twoxili (talk) 20:15, 1 June 2023 (UTC)

For posterity, the statement "the hypothetical bottom row of Pascal's triangle in base n is, on the first order of magnitude, e" is just a fancy way of stating the limit definition of e, which is why its noteworthiness in the below discussion is dubious. Twoxili (talk) 14:11, 5 June 2023 (UTC)

I'll try adding that parse of your statement: "the hypothetical bottom row of Pascal's triangle in base n is, on the first order of magnitude, e" if it helps clarify the point of interest. Twoxili (talk) 23:24, 1 June 2023 (UTC)

I guess my opinion right now is that this limit is not notable. It's a nice math fact, but not all nice math facts need to be on Wikipedia. (For example, no theorem that I have professionally proved is on Wikipedia.) I could be convinced otherwise, if a few Wikipedia:Reliable sources were marshaled, to confirm that this limit is important, commonly used, or especially interesting. Mgnbar (talk) 23:32, 1 June 2023 (UTC)

I'll look at what's out there, and if I can't find anything, then I'll have to concur. Twoxili (talk) 23:43, 1 June 2023 (UTC)

Does the broad interest in the $2^{n}$ row sums and the $11^{n}$ row values not imply a similar level of interest in the shared mechanism behind these two phenomena? What I mean is, the reason why these two cases are generalizable across bases is due to the fact that the rows converge to $e$ when $a=n$. The base of the natural log is the mathematical "glue" that connects the bases and gives their powers identical numerical expressions. This was Prof. Morton's point, but he didn't say it in these terms (apart from including "11" in the title of his article), and for that reason I think the point has been overlooked. We've been discussing the implications of this principle but not the principle itself. I've updated the intro of this thread to propose this description replace the The value of a row bullet point under the Rows section. * This has been edited Twoxili (talk) 10:35, 2 June 2023 (UTC)

The divisibility rule for 11 is generalizable to $11_{n}$. This is a corollary of Morton's paper and a couple of follow-up papers by author's other than Morton were written on this topic (I'll add these citations [Edit: the total is 6 now compared to the 1 citation in the earlier revision]). That further establishes the relevance of $11$ to discussions about the triangle or binomials in general. When we make statements about $11$, we're making statements about the root of $e$, which is the common thread through the bases that makes this generalization possible. Twoxili (talk) 14:44, 2 June 2023 (UTC)

When we make statements about 11, we're making statements about the root of e You keep repeating versions of this statement, and I am perfectly happy to accept that you believe it strongly. But Wikipedia is based on what reliable sources write, and until there is evidence that reliable sources explicitly make this connection, I don't think you're going to find a lot of support to include it in the article. (Stackexchange sites, which you mention below, are WP:USERGEN and so generally unacceptable.) --JBL (talk) 17:17, 2 June 2023 (UTC)

I also agree that the connection is not explicitly asserted in the paper. And I do realize the importance of keeping original content out of the articles. I agree Morton and the other authors who wrote follow up papers might not have realized they were talking about $e$. I'm not claiming they did. My question, though, is if an author uses the Leibniz formula for $\pi$, or the Euler formula, rather than the symbol $\pi$, and makes statements about those formulas, is it not okay for the Wikipedia article on $\pi$ to reference the paper, and replace those formulas with the symbol $\pi$ when paraphrasing the paper, if it helps get the points of the paper across to an audience interested in $\pi$? Alternatively, if I remove all instances of the symbol $e$, and replaced them with expressions involving $1.1_{n}$, will the edit be acceptable? because the content would not change as a result of that refactoring. The limit of $1.1_{n}^{n}=(1+{\frac {1}{n}})^{n}$ is the constant $e$ as $n$ approaches $\infty$. This is a natural language description of the limit equation of $e$. There are many more descriptions of $e$. The limit of $11_{n}^{n}$ is the last row of the base-$n$ triangle as $n$ approaches $\infty$. That is the binomial theorem as applied by Morton. The former description is on the first order of magnitude of base $n$, and the latter is on the last order of magnitude of base $n$. Morton's paper, by choosing to discuss 11 in arbitrary bases (radix $n$) and its powers as rows of Pascal's triangle (which is unbounded at its base), is tantamount to a discussion about $e$. * This has been edited to correct several errors in describing e and Morton's use of the binomial theorem Twoxili (talk) 17:33, 2 June 2023 (UTC)

I do not think your analogy is even slightly apt. Rather than try to force your (idiosyncratic, to say the least) views into this article, you should try to improve Wikipedia by making sure it reflects accurately what the best-quality sources have to say on the subject. Otherwise, you run a significant risk of becoming disruptive. --JBL (talk) 17:51, 2 June 2023 (UTC)

I greatly appreciate your feedback. Twoxili (talk) 17:53, 2 June 2023 (UTC)

I've removed all instances of e, and changed the language to focus on explaining the paper faithfully and succinctly. Does the revised version come any closer to resolving your concerns? Twoxili (talk) 22:19, 2 June 2023 (UTC)

So far I was able to find this math.stackexchange.com post. I realize this may not be a standard source, but it is an example of what I think is a big problem about this topic. Here, the OP discovered this property in reverse, but never got an answer as to what he discovered. He appears to have started by looking at $(1.1)^{n}$, and realized the numbers match the Pascal rows with decimals, and asks about the relationship of $e$ to the triangle (he even cites the row product ratio connection as something he wanted to know more about, just as I did). He offers a conjecture that turns out to be incorrect. The accepted answer correctly explains that the binomial $a^{n}\left(1+{\frac {1}{a}}\right)^{n}=11_{a}^{n}$ is the culprit, but not in those terms. The answer fails to explain why $a^{n}$ is essential for the convergence to work, and instead demonstrates arithmetically why OP's approach doesn't work. According to my writeup, the OP's product is off because $e$ cross-cuts the sequence of radix-$a$ triangles at row $n$, and moving off that plane/triangle will move you away from $e$: "If it drops faster or slower, the limit will be smaller than or larger than $e$" the answer explains in describing the asymmetry of OP's scaling, but what I wish to draw attention to is how it goes about explaining it. The answer makes no mention at all of radices. It didn't come up in the comments either. And no one comments that OP is partially correct in seeking a formula for $e$ in the Pascal rows: "Is there any way to make this series converge to $e$ with the adding of zeroes suitably as needed?" Yes, but you need to think of $e$ as moving across bases forever towards its limit. The OP posits "$n$-many $0$s" between $a=1$ and $b=1/n$, so he intuits the scaling of $n$ to the power $n$ (he neglects to scale the other term, $a$, of the binomial), but the answer simply responds in binary. The answer mentions another indication that you're off the plane $e$: "... then even in the limit the first digit (the only digit before the decimal point) is always $1$, which is the left edge of the triangle", but does not mention that a first digit of $2$ nevertheless holds for all rows $a=n$ for $n>2$ in base $n$ (because these rows are indeed converging to $e$). In my writeup, I note this also. The last digit of OP's product is $1$ because the penultimate $1$ will not carry in the normalized value of the row in OP's scheme. That 1 is one of the four a priori values in the limit. If you don't know this, then the triangle's leftmost diagonal will mislead you to believe the convergence is not possible. I myself nearly dismissed the idea because of this, and OP actually does: "Yeah I guess that was a blunder on my side. I guess there is indeed no way." I also show that one application of this fact is that we know the exact length of the partial product's numerical value. The reply appears to do a nice job of explaining why the OP's conjecture doesn't converge to $e$ (not all of which I understand after my cursor look at it, I admit), but fails to mention the useful fact that the rows are, in fact, partial products of $e$. More to the point, I believe the OP would have appreciated and might have been enriched by a more thorough explanation of the intricate relationship between $e$ and the hypothetical bottom row. I think my writeup is what the OP might have been looking for, or would have at least found interesting. The post has 779 views, so it seems he and I are not the only ones interested in this relationship. I'll continue searching (scholarly searches are turning up nothing so far), but do you think this post lends any credit to why it's appropriate to add this to the Wikipedia page? * This has been edited Twoxili (talk) 01:29, 2 June 2023 (UTC)

Here is another post that get's the row formula's structure right (but fatally fails to equate $n$ and $k$). I can't tell by "make's sense at all" whether OP asks how to normalize the the $k$th row numeral or if the question is whether the presented formula describes the $k$th row, but his second question seems to ask whether the rows have a limit. As stated, his conjecture does not converge. The post has no answers, but over 100 views. Twoxili (talk) 11:51, 2 June 2023 (UTC)

Since we know that the ratio of the ratio of successive row (coefficient) products converge to $e$, and we also have a definition of the row that converges to $e$, the rows are partial products of the product ratio growth. That connection could be useful in understanding the product ratio growth, a topic that is widely of interest. Twoxili (talk) 03:04, 2 June 2023 (UTC)

First sentence

In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra.

or

In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arise in probability theory, combinatorics, and algebra.

i.e., is "arise" plural or singular? User:Maryanne Cunningham recently changed "arises" to "arise", and User:JayBeeEll reverted. I'd say both are possible - the triangle arises, but the coeffficients also arise. I think the "arise" version is marginally easier to read - and, in fact, you could say that it is the coefficients that arise in the fields mentioned, where as the triangle is merely a (slightly quirky) way of displaying them. Nø (talk) 19:09, 19 June 2023 (UTC)

The subject of both the verb "arises" and the article is undoubtedly Pascal's triangle. To make the subject of the verb be "binomial coefficients", one would write, "Pascal's triangle is a triangular array of binomial coefficients, which arise in ..." -- but since the article is about Pascal's triangle, it would be odd to have the first sentence be about some other object. (I agree with you that the sentence structure is slightly difficult to parse, even for native speakers.) --JBL (talk) 20:18, 19 June 2023 (UTC)

How about

In mathematics, Pascal's triangle is a triangular array of the binomial coefficients arising in probability theory, combinatorics, and algebra.

It circumvents the question, and is shorter. I (a nonnative speaker) find that it is perhaps marginally easier to parse. Nø (talk) 07:57, 23 June 2023 (UTC)

@Nø: Yes that seems reasonable. --JBL (talk) 17:56, 23 June 2023 (UTC)

Done. Nø (talk) 12:53, 25 June 2023 (UTC)

Extension to negative rows and columns, and other fringe stuff

The lengthy section Pascal's triangle#To the integers, extending the triangle upwards to the left and right, relies on a single source, and I think it has undue weight, as it is of very marginal interest. Briefly stated,

Pascal's triangle may be extended upwards, above the 1 at the apex, preserving the additive property, but there is more than one way to do so, and it is rarely of any use.

I suggest replacing the whole section by this statement, perhaps with the same source cited at the last comma, and possibly with a a link to a separate article containing the present section. Thoughts?

Other sections, including Pascal's triangle#Calculating a row or diagonal by itself (which I rewrote some time ago, as far as I recall making it quite a bit longer, but more readable), also seem to be of marginal interest only, making this article overly long and much less accessible than it could be. Thoughts on that? Nø (talk) 11:48, 27 June 2023 (UTC)

Having not surveyed the literature myself, but having only read this article, I agree with you. In your proposed sentence, we should remove the clause "and it is rarely of any use", which verges on speculation and editorial.

Similarly, I think that the recently added section "To arbitrary bases" is overly long and detailed. Mgnbar (talk) 12:15, 27 June 2023 (UTC)

This was discussed above at #Extending Pascal's triangle and I think basically everyone agrees that section is excessive but no one has done anything about it :). --JBL (talk) 17:35, 27 June 2023 (UTC)

Done. How about the section on calculating a row or column by itself? I think it is less "sometinh someone cooked up one afternoon after class", but still I'm not sure its presense makes the article better. Thoughts on that, or other sections?

Also, is there a neat way to split the article into one shorter and more accessible article , and another picking up on more periferal or advanced topics? The latter would by nature be very fragmented (or it would be a ridiculous number of very short articles). Nø (talk) 12:37, 29 June 2023 (UTC)

Proportion of odd coefficients

A recent edit added, with citation: "As the proportion of black numbers tends to zero with increasing n, a corollary is that the proportion of odd binomial coefficients tends to zero as n tends to infinity." In trying to clean this up, I realized that I didn't know what "proportion" meant exactly.

Let O(n) be the number of odd binomial coefficients in row n, and let E(n) be the number of even ones.

It is not true that $\lim _{n\to \infty }O(n)/(O(n)+E(n))=0$, because all coefficients in row n are odd when $n=2^{k}-1$.

So is the claim instead that

$\lim _{n\to \infty }{\frac {\sum _{i=1}^{n}O(i)}{\sum _{i=1}^{n}O(i)+E(i)}}=0?$

Mgnbar (talk) 12:06, 12 September 2023 (UTC)

Mention Jia Xian triangle?

We probably should mention the Jia Xian triangle here as well. The Page Jia_Xian#Biography says about this relationship: > Jia Xian described the Pascal's triangle (Jia Xian triangle) around the middle of the 11th century, about six centuries before Pascal. Jia used it as a tool for extracting square and cubic roots

This appears to be a kinda important historical fact and probably should be included here as well. Agowa (talk) 01:22, 10 March 2024 (UTC)

Jia Xian is already mentioned in the history section of this article. According to what is written there, his name is not usually attached to it, even in China; do you have any sources that support the contrary view? (The only source in Jia Xian#Biography is not available online as far as I can see, so I am not able to immediately check what is written there.) --JBL (talk) 23:27, 10 March 2024 (UTC)