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 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 nth row equals the middle element of the 2nth."

But it appears to me that the claim should be "the sum of the squares of the elements of the nth row equals the middle element of the 2nth-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 &mdash; 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 nth 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 nth 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 nth row numbering starts with the 0th row.
 * In the first section, there is also some confusion in that the page states "for positive integers n and k where n &ge; 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 &ge; k", with all the math formulas based on n starting at zero, and the text should be updated to not use the phrase "nth row", but something more appropriate.  Either the text should be changed to indicate that we consider the first row to be the 0th row, or use another variable (like m) to indicate the row number and give it's relationship to n.  Or just change nth 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 . 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 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)

Extension to negative rows and columns, and other fringe stuff
The lengthy section Pascal's triangle, 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 (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 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 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 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)