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2730:. It is now a year and a half later, and no merge has occurred. Both articles are long, almost eye-wateringly long; making an even larger mega-article seems wrong. The tradition in WP is that grand generalizations of some concept get their own article. FWIW, currently, neither article hints at what the generalization is. The commentary above suggests that its some
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n n lim ( Σ k) / { ( Σ k). n } = 1/2 n→∞ k=1 k=1 n n lim ( Σ k) / { ( Σ k). n } = 1/3 n→∞ k=1 k=1 n n lim ( Σ k) / { ( Σ k). n } = 1/4 n→∞ k=1 k=1 n n lim ( Σ k) / { ( Σ k). n } =
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says that articles should be listed under their common names: " names and terms commonly used in reliable sources, and so likely to be recognized...Articles are normally titled using the most common
English-language name of the subject of the article." This holds even when that name is erroneous or
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What I would like to add next is a generalisation to non-natural powers which shows the relationship between
Faulhauber's formula and the Hurwitz (and Riemann) zeta function, and further demonstrates why the alternate expression is more natural (since the more conventional one is not a Taylor series
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to be +1/2; the other has −1/2. That's why they don't contradict each other. Look at the polynomials listed in this article, and you see that in each case if you change the coefficient of the second-highest-degree term from plus to minus, the effect is exactly the same as that of deleting the last
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I believe what I've written so far is explicative rather than merely demonstrative, since the proof is in a historic style, illuminating the relation between coefficients that led to the formulation of the
Bernoulli numbers, and the transformation between the two expressions explains why there are
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The complaints in the tag make sense, as the notation for failing factorial is not of a common use. So, not recalling it suppose that the reader of this section has memorized the lead. As one cannot impose such an effort to the reader, the tag is justified. For the change of notation, this is the
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and frankly, I do not understand why that topic is notable and worthy of an article on WP. It seems to describe dull, obvious generalizations that you can cook up in an afternoon or two of scratching around; why is the topic notable? Am I missing something? There don't seem to be any theorems or
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without being interested in the famous formula would lose the opportunity to deepen unless they also incorporate the same article, appropriately modified, in them. It seems to me a useless waste of energy that could be used much better for example by explaining with adequate sources that strange
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There is a complaint about a notation not explained, which is explained in the first few lines of the article. There is a complaint about an index name change, which is futile. Nobody able to read this article would be confused by this and it draws attention to the change of the lower limit. Of
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2711:. I am a random visitor from the Internet, with other words, one of your users. I find the idea of the merging simply hilarious - I googled for Faulhabers' formula and I got exactly what I wanted. You want now eliminate this and merge into a cloudy, similar thing. Do not do that!
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There is an error in faulhaber's polynomials. This article says that : 1^{11}+ 2^{11} + 3^{11} + \cdots + n^{11} = {32a^6 - 64a^5 + 68a^4 -40a^3 + 5a^2 \over 6} But It's : 1^{11}+ 2^{11} + 3^{11} + \cdots + n^{11} = {32a^6 - 64a^5 + 68a^4 -40a^3 + 10a^2 \over 6} Thank you.
1233:, because that theorem relates the differences of terms in a sequence to the terms, and the total of a summation is itself a sequence indexed by the upper limit of the summation, where the difference between one total and the "next" is the next term of summation.
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Everyone in this discussion understand the (essentially trivial) way in which the topic of the new article generalizes the topic of this article. Given the relative importance of the generalization, it is best treated in a brief section of this article.
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These identities follow trivially from the
Riemann integral of x^p between 0 and 1 evaluated as a limit with a regular dissection of width 1/n. The factor of 2 arising in the formula because the denominator is ½(n+1)n^p rather than just n^{p+1}.
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which, by induction, can be generalized in a formula. Note that the denominators of the results are the positions of the exponents in the numerator in the sequence of odd numbers. Also this is a special case? To see the genesis of this, go to:
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The proof I had written avoids calculus, generating functions, and even glosses over an induction step, in the hope that it would be an easier "entry point" for those wishing to get an idea of the significance of the
Bernoulli numbers.
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Even if from a historical point of view it seems the opposite, the traditional
Faulhaber formula identifies a countable infinity of polynomials calculating sums of powers but this is only a point of the four-dimensional complex space
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This new article covers substantially the same substantive material, from a slightly different perspective. There is no meaningfully separate study of the sums of powers of arithmetic progressions in general and the study of the sum
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I deliberately chose a very elementary proof as the first version, partly because
Faulhaber's formula seems to be the historical origin of the Bernoulli numbers; i.e., formula first, Bernoulli numbers later.
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I have more material I would like to add to this page, but I don't want to do it without gauging whether it is in the spirit of
Knowledge, since I'm aware that proof-like material is not quite encyclopaedic.
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is trivial but the reflections on the downsizing of the claims of the point-formula in a much vaster four-dimensional complex universe are not trivial. Also in the wikipedia universe a similar reflection is
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or something like that, but nothing to that effect is mentioned in either article. What is this variety that User:bio.oid speaks of? Where are these "points in general position"? ... Hmm. I keep looking at
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1812:'s definition of the overline is correct, this should be more clearly mentioned in the article. However, this would still not answer the first question which needs to be answered: what is the definition of
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Unless it appears that the term "Bernoulli's formula" is more common than "Faulhaber's formula", the article should be titled "Faulhaber's formula", perhaps with "Bernoulli's formula" as a redirect to it.
1474:. His proof is linked in each contribution. The previous sequences, obtained in a different manner, were already present in the database. Later ones were blocked shots by editors, for practical reasons.
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A letter from Knuth informs me that in the paper version of his article the mistake is not there. So probably
Faulhaber had it right and it was just a misprint in the preprint that was later corrected.
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The papers present two types of iterated sums called recurrent sums and multiple sums. Then present the generalizations for recurrent and multiple sums (See
Theorem 4.8 of and Theorem 5.1 of ).
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A meta remark. It requires courage and expertise to remove that remark, both of which I lack. Why are those remarks anonymous? I would like to file a complaint of vandalism against such author.
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2. El Haddad, R. (2022). A generalization of multiple zeta values. Part 2: Multiple sums. Notes on Number Theory and Discrete Mathematics, 28(2), 200-233, DOI: 10.7546/nntdm.2022.28.2.200-233.
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1. El Haddad, R. (2022). A generalization of multiple zeta value. Part 1: Recurrent sums. Notes on Number Theory and Discrete Mathematics, 28(2), 167-199, DOI: 10.7546/nntdm.2022.28.2.167-199.
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But neither article mentions the complex numbers or use the word "variety". All coefficients appear to be integers or rationals, not complex numbers. I imagine this theory generalizes to
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Lots of things in math aren't named after the person to actually discover them. We don't need to remind the reader at every opportunity, and certainly don't need the commentary.
554:{\displaystyle \sum _{k=1}^{n}k^{p}={1 \over p+1}\sum _{j=0}^{p}{p+1 \choose j}B_{j}n^{p+1-j}\qquad \left({\mbox{with }}B_{1}={1 \over 2}{\mbox{ rather than }}-{1 \over 2}\right)}
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In short, it will be my limit, but I can only see negative aspects in this merger project but it seems to me that the positive ones have not yet been explained. With wiki love --
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Hello - there seems to be an error in the formula for 9th powers in the image "Jakob Bernoulli's Summae Potestatum, Ars Conjectandi, 1713". The formula for 9th powers should be
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865:"Pell's equation", even though Pell had nothing to do with it, the equation was solved by Brouncker, and the solution was attributed to Pell entirely by mistake. Similarly
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In my opinion, this is a possible in-depth study that must be well connected but remain separate. Also because the same insights must be available coming, for example, from
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Setting n=1 makes it clear that there was an error, and I have changed it according to what you say. However, Knuth's paper does say 5n^2. I don't know whose mistake it is.
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A special case is still needed for p=0 though, because the formula right now gives 1+1+...1 = n+1/2, which is obviously wrong. 10:33, 21 February 2019 (UTC)
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same: It is easy to verify it, but it is a task for the editor, not for the reader. Nevertheless, I have modified the formulation, and removed the tag.
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be inserted, all is well - since all odd Bernoulli numbers (excepting B_1) vanish. (There should be no other consequences - as far as I can see.)
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Examples of particular cases of the generalization Faulhaber's formulas are present in the OEIS (some of which are cited in the two papers).
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2951:{\displaystyle {\frac {1}{10}}n^{10}+{\frac {1}{2}}n^{9}+{\frac {3}{4}}n^{8}-{\frac {7}{10}}n^{6}+{\frac {1}{2}}n^{4}-{\frac {3}{20}}n^{2}.}
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No, the formula where it appears is its definition. The meaning of the overline is somehow explained in the following text: one pass from
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The subject of the newly created voice is the sets of particular polynomials that populate this vast space, not the historical formula.
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Different expressions require different conventions; is there any part of the page where it is ambiguous which convention is needed?
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2664:" conflicts with other articles that would be deprived of it. This is because those who are interested in articles such as
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Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.
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What is a reference for the formula which follows "More generally," in the Faulhaber polynomials section of the article?
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http://en.wikipedia.org/Talk:Summation#Sum_of_the_first_.22n.22_cubes_-_even_cubes_-_odd_cubes_.28geometrical_proofs.29
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A generalization Faulhaber's formula for two types of iterated sums is presented in the following two papers:
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The following Wikimedia Commons file used on this page or its Wikidata item has been nominated for deletion:
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By the way, this tag is not anonymous, you can find its its author by consulting the history of the article.
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Is a proof based on an exponential generating function for Bernoulli polynomials appropriate for this page?
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This does not appear a merge operation but a crude attempt at transplantation. What is your opinion?. --
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Since the formula is not actually due to Faulhaber, should we move the article to "Bernoulli's formula"?
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I suggest adding these formulas to the Faulhaber's formula page as it is very relevant to the topic.
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by changing the signs of the elements such that the sum of the column index and the row index is odd (
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appears without prior use or definition. Perhaps it was introduced in an earlier, now-deleted section.
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can be used to calculate the second partial sums of m-th powers, with the following general formula: a
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I seem to recall that he knew the first couple of dozen or so cases. I'll see if I can find that.
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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is not listed as "Kuratowski's lemma", even though Kuratowski unquestionably published it first.
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In a 1631 edition of Academiae Algebrae, J. Faulhaber published the general formula for...
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Yes; it might be fair to say that most limits involving summations are a special case of
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No, Faulhaber did definitely not know this formula. It is absurd to call this formula
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Bernoulli polynomial with + exponent that is encountered in the cannibal article :-).
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A Commons file used on this page or its Wikidata item has been nominated for deletion
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Error in image taken from Jakob Bernoulli's Summae Potestatum, Ars Conjectandi, 1713
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In the 'Umbral' section it looks as if the factor 1 / (p+1) has been omitted twice.(
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What information is correct? Did Faulhaber know (and publish) the general formula?
2429:. I agree that this does not seem sufficiently distinct for a separate article. —
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course now the formula is more in agreement what is customary in the textbooks.
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are undefined in the current version of the article; cleanup is indeed needed.
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When it was a draft, I commented that this was essentially a content fork of
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Performing calculations with Excel, I encounter these other amazing results:
715:{\displaystyle \sum _{k=0}^{m-1}k^{n}=0^{n}+1^{n}+2^{n}+\cdots +{(m-1)}^{n}}
2565:{\displaystyle (1,1)\in \mathbb {C} ^{2}\quad {\text{but not vice versa}}}
815:. Mathworld is, again, unreliable and cannnot be used a primary source.
1446:. This formula generated seven new contributions to the OEIS database:
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Unknown matrix variable in "From examples to matrix theorem" section
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claims being made. Is there a reason not to prod or AfD the thing?
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Polynomials calculating sums of powers of arithmetic progressions
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Polynomials calculating sums of powers of arithmetic progressions
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Polynomials calculating sums of powers of arithmetic progressions
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Please see at point 2 of Discussion in the following article:
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https://commons.wikimedia.org/File:Pubblicazione_english_01.pdf
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Comments on new sections? ("Proof" and "Alternate expression")
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In this article, in the first formula, there is no need for a
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Thank you. I have now added text to make things more clear.
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The formula's above don't make sense, but it looks as if
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1712:{\displaystyle \textstyle a_{i,j}\to (-1)^{i+j}a_{i,j}}
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about faulhaber observation in the last of this article
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2281:Participate in the deletion discussion at the
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1426:A practical use of the Faulhaber’s formulas.
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321:article? The upper limits differ by 1.--
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740:term on the left side of the identity.
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2201:2602:306:CC10:5D70:B857:4E4E:215A:FE59
3006:oppure in questa stessa discussione:
2276:Pascal’s and Bernoulli’s triangle.jpg
1502:Faulhaber polynomials citation needed
1192:n). n = 1/4 n→∞ 1 n lim (Σ
1184:n). n = 1/3 n→∞ 1 n lim (Σ
1176:n). n = 1/2 n→∞ 1 n lim (Σ
1142:. It is interesting though, isn't it?
317:Where is the mistake? Here on in the
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3064:Implementation of the merger project
2315:The following discussion is closed.
95:This article is within the scope of
2736:"...four-dimensional complex space
2029:{\displaystyle {\overline {A}}_{m}}
1941:{\displaystyle {\overline {A}}_{m}}
1907:{\displaystyle {\overline {A}}_{m}}
1787:{\displaystyle {\overline {A}}_{m}}
1633:{\displaystyle {\overline {A}}_{m}}
1560:{\displaystyle {\overline {A}}_{m}}
38:It is of interest to the following
2960:The last term seems to be wrong.
2036:but with all positive elements". —
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3097:Low-priority mathematics articles
1570:Previous comment was coming from
115:Knowledge:WikiProject Mathematics
2765:where each point does the same."
2758:{\displaystyle \mathbb {C} ^{2}}
2699:The discussion above is closed.
2507:{\displaystyle \mathbb {C} ^{2}}
1574:at 16 October 2017, time 04:27h.
118:Template:WikiProject Mathematics
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2514:where each point does the same.
1514:) 17:07, 28 January 2017 (UTC)
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857:misattributed. For example,
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109:and see a list of open tasks.
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2728:Don't merge; maybe delete?
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1430:The Faulhaber’s formulas F
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212:{\displaystyle {(-1)^{p}}}
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2660:The will to incorporate "
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1410:I second your opinion. --
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939:21:25, 13 June 2012 (UTC)
918:when p is non-natural).
914:two conventions for B_1.
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2701:Please do not modify it.
2318:Please do not modify it.
2215:clarification not needed
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1200:n). n = 1/5 n→∞ 1
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305:09:16, 14 May 2010 (UTC)
141:project's priority scale
2670:arithmetic progressions
2579:arithmetic progressions
2299:
229:I have changed it now.
98:WikiProject Mathematics
2952:
2759:
2650:
2566:
2508:
2405:
2187:
2156:
2125:
2098:
2030:
1996:
1969:
1942:
1908:
1874:
1833:
1788:
1754:
1713:
1634:
1600:
1561:
1258:means the following:
1136:
1040:
768:claims the opposite:
716:
629:
555:
426:
374:
213:
28:This article is rated
2953:
2760:
2674:Bernoulli polynomials
2651:
2567:
2509:
2406:
2188:
2186:{\displaystyle -3/20}
2157:
2155:{\displaystyle -1/12}
2126:
2124:{\displaystyle n^{2}}
2099:
2031:
1997:
1995:{\displaystyle A_{m}}
1970:
1968:{\displaystyle G_{m}}
1943:
1909:
1875:
1873:{\displaystyle A_{m}}
1834:
1832:{\displaystyle A_{m}}
1789:
1755:
1753:{\displaystyle A_{m}}
1714:
1635:
1601:
1599:{\displaystyle A_{m}}
1562:
1137:
1041:
717:
603:
556:
406:
354:
214:
2807:
2740:
2613:
2522:
2489:
2356:
2166:
2135:
2108:
2082:
2006:
1979:
1952:
1918:
1884:
1857:
1816:
1764:
1737:
1644:
1610:
1583:
1537:
1231:Stolz–Cesàro theorem
1050:
979:
973:Stolz–Cesàro_theorem
756:did or did not know?
600:
351:
185:
181:for p=1: because if
121:mathematics articles
2465:Faulhaber's formula
2446:Makes sense to me.
2306:Faulhaber's formula
2097:{\displaystyle p=9}
813:Faulhaber's formula
732:This article takes
588:The other one says:
2988:Yes, it is wrong.
2948:
2755:
2646:
2562:
2559:but not vice versa
2556:
2504:
2401:
2330:consensus to merge
2300:Proposed merge of
2287:Community Tech bot
2183:
2152:
2121:
2094:
2026:
1992:
1965:
1948:as the inverse of
1938:
1904:
1870:
1829:
1784:
1750:
1709:
1708:
1630:
1596:
1557:
1345:Ice ax1940ice pick
1313:Ice ax1940ice pick
1132:
1036:
712:
551:
531:
501:
490:
313:Upper limit of sum
209:
90:Mathematics portal
34:content assessment
2966:comment added by
2933:
2910:
2887:
2864:
2841:
2818:
2732:algebraic variety
2560:
2241:
2229:comment added by
2211:
2199:comment added by
2018:
1930:
1896:
1776:
1622:
1575:
1549:
1498:
1481:comment added by
1360:
1343:comment added by
1328:
1311:comment added by
942:
925:comment added by
760:The article says
544:
530:
524:
500:
449:
404:
276:
262:comment added by
155:
154:
151:
150:
147:
146:
3104:
2982:
2957:
2955:
2954:
2949:
2944:
2943:
2934:
2926:
2921:
2920:
2911:
2903:
2898:
2897:
2888:
2880:
2875:
2874:
2865:
2857:
2852:
2851:
2842:
2834:
2829:
2828:
2819:
2811:
2773:ring of integers
2769:integral domains
2764:
2762:
2761:
2756:
2754:
2753:
2748:
2655:
2653:
2652:
2647:
2627:
2626:
2621:
2571:
2569:
2568:
2563:
2561:
2558:
2554:
2553:
2548:
2513:
2511:
2510:
2505:
2503:
2502:
2497:
2410:
2408:
2407:
2402:
2400:
2399:
2381:
2380:
2368:
2367:
2320:
2192:
2190:
2189:
2184:
2179:
2162:. It should say
2161:
2159:
2158:
2153:
2148:
2130:
2128:
2127:
2122:
2120:
2119:
2103:
2101:
2100:
2095:
2035:
2033:
2032:
2027:
2025:
2024:
2019:
2011:
2001:
1999:
1998:
1993:
1991:
1990:
1974:
1972:
1971:
1966:
1964:
1963:
1947:
1945:
1944:
1939:
1937:
1936:
1931:
1923:
1913:
1911:
1910:
1905:
1903:
1902:
1897:
1889:
1879:
1877:
1876:
1871:
1869:
1868:
1838:
1836:
1835:
1830:
1828:
1827:
1793:
1791:
1790:
1785:
1783:
1782:
1777:
1769:
1759:
1757:
1756:
1751:
1749:
1748:
1718:
1716:
1715:
1710:
1707:
1706:
1691:
1690:
1663:
1662:
1639:
1637:
1636:
1631:
1629:
1628:
1623:
1615:
1605:
1603:
1602:
1597:
1595:
1594:
1569:
1566:
1564:
1563:
1558:
1556:
1555:
1550:
1542:
1497:
1475:
1359:
1337:
1334:Bernoulli number
1327:
1305:
1141:
1139:
1138:
1133:
1131:
1130:
1112:
1111:
1081:
1080:
1068:
1067:
1045:
1043:
1042:
1037:
1035:
1034:
1010:
1009:
997:
996:
941:
919:
721:
719:
718:
713:
711:
710:
705:
678:
677:
665:
664:
652:
651:
639:
638:
628:
617:
571:runs only up to
560:
558:
557:
552:
550:
546:
545:
537:
532:
528:
525:
517:
512:
511:
502:
498:
488:
487:
466:
465:
456:
455:
454:
445:
433:
425:
420:
405:
403:
389:
384:
383:
373:
368:
343:The formula says
319:Bernoulli_number
275:
256:
218:
216:
215:
210:
208:
207:
206:
123:
122:
119:
116:
113:
92:
87:
86:
76:
69:
68:
63:
55:
48:
31:
25:
24:
16:
3112:
3111:
3107:
3106:
3105:
3103:
3102:
3101:
3082:
3081:
3066:
3030:
2961:
2935:
2912:
2889:
2866:
2843:
2820:
2805:
2804:
2799:
2743:
2738:
2737:
2705:
2704:
2616:
2611:
2610:
2543:
2520:
2519:
2492:
2487:
2486:
2391:
2372:
2359:
2354:
2353:
2349:
2316:
2309:
2283:nomination page
2269:
2217:
2164:
2163:
2133:
2132:
2111:
2106:
2105:
2080:
2079:
2076:
2009:
2004:
2003:
1982:
1977:
1976:
1955:
1950:
1949:
1921:
1916:
1915:
1887:
1882:
1881:
1860:
1855:
1854:
1819:
1814:
1813:
1767:
1762:
1761:
1740:
1735:
1734:
1692:
1676:
1648:
1642:
1641:
1613:
1608:
1607:
1586:
1581:
1580:
1540:
1535:
1534:
1531:
1504:
1476:
1445:
1441:
1437:
1433:
1428:
1383:
1338:
1306:
1303:
1298:
1263:
1201:
1199:
1195:
1191:
1187:
1183:
1179:
1175:
1171:
1116:
1097:
1072:
1053:
1048:
1047:
1026:
1001:
982:
977:
976:
948:
920:
907:
888:
859:Pell's equation
835:
758:
738:
688:
669:
656:
643:
630:
598:
597:
503:
495:
491:
467:
457:
435:
428:
393:
375:
349:
348:
315:
257:
252:
198:
183:
182:
175:
160:
120:
117:
114:
111:
110:
88:
81:
61:
32:on Knowledge's
29:
12:
11:
5:
3110:
3108:
3100:
3099:
3094:
3084:
3083:
3070:37.161.247.176
3065:
3062:
3029:
3026:
3025:
3024:
3023:
3022:
3004:
2947:
2942:
2938:
2932:
2929:
2924:
2919:
2915:
2909:
2906:
2901:
2896:
2892:
2886:
2883:
2878:
2873:
2869:
2863:
2860:
2855:
2850:
2846:
2840:
2837:
2832:
2827:
2823:
2817:
2814:
2798:
2795:
2794:
2793:
2752:
2747:
2724:
2723:
2698:
2697:
2696:
2695:
2694:
2693:
2692:
2678:
2658:
2645:
2642:
2639:
2636:
2633:
2630:
2625:
2620:
2575:
2572:
2552:
2547:
2542:
2539:
2536:
2533:
2530:
2527:
2516:
2515:
2501:
2496:
2479:
2458:
2441:
2431:David Eppstein
2398:
2394:
2390:
2387:
2384:
2379:
2375:
2371:
2366:
2362:
2348:
2347:
2346:
2345:
2344:
2311:
2310:
2308:
2298:
2279:
2278:
2268:
2265:
2264:
2263:
2248:
2247:
2216:
2213:
2182:
2178:
2174:
2171:
2151:
2147:
2143:
2140:
2118:
2114:
2093:
2090:
2087:
2075:
2072:
2071:
2070:
2069:
2068:
2067:
2066:
2065:
2064:
2063:
2062:
2038:David Eppstein
2023:
2017:
2014:
1989:
1985:
1962:
1958:
1935:
1929:
1926:
1901:
1895:
1892:
1867:
1863:
1826:
1822:
1781:
1775:
1772:
1747:
1743:
1705:
1702:
1699:
1695:
1689:
1686:
1683:
1679:
1675:
1672:
1669:
1666:
1661:
1658:
1655:
1651:
1627:
1621:
1618:
1593:
1589:
1568:
1554:
1548:
1545:
1530:
1527:
1503:
1500:
1483:Ancora Luciano
1443:
1439:
1435:
1431:
1427:
1424:
1423:
1422:
1382:
1379:
1378:
1377:
1376:
1375:
1301:
1297:
1294:
1293:
1292:
1260:
1256:Ancora Luciano
1252:
1251:
1250:
1249:
1248:
1247:
1234:
1212:Ancora Luciano
1197:
1193:
1189:
1185:
1181:
1177:
1173:
1169:
1167:
1166:
1165:
1164:
1163:
1157:
1156:
1143:
1129:
1126:
1123:
1119:
1115:
1110:
1107:
1104:
1100:
1096:
1093:
1090:
1087:
1084:
1079:
1075:
1071:
1066:
1063:
1060:
1056:
1033:
1029:
1025:
1022:
1019:
1016:
1013:
1008:
1004:
1000:
995:
992:
989:
985:
958:Ancora Luciano
947:
946:A new Theorem?
944:
906:
903:
887:
884:
883:
882:
870:
834:
831:
830:
829:
828:
827:
806:
805:
776:
775:
757:
754:
753:
752:
736:
730:
728:
727:
726:
725:
724:
723:
722:
709:
704:
701:
698:
695:
692:
687:
684:
681:
676:
672:
668:
663:
659:
655:
650:
646:
642:
637:
633:
627:
624:
621:
616:
613:
610:
606:
589:
586:
584:
583:
582:
581:
580:
565:
563:
562:
561:
549:
543:
540:
535:
523:
520:
515:
510:
506:
494:
486:
483:
480:
477:
474:
470:
464:
460:
453:
448:
444:
441:
438:
432:
424:
419:
416:
413:
409:
402:
399:
396:
392:
387:
382:
378:
372:
367:
364:
361:
357:
344:
337:
314:
311:
310:
309:
308:
307:
264:63.243.163.199
251:
248:
247:
246:
245:
244:
221:Hair Commodore
205:
201:
197:
194:
191:
174:
171:
159:
158:Umbral section
156:
153:
152:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
3109:
3098:
3095:
3093:
3090:
3089:
3087:
3080:
3079:
3075:
3071:
3063:
3061:
3060:
3056:
3052:
3048:
3045:
3042:
3039:
3036:
3033:
3027:
3021:
3017:
3013:
3009:
3005:
3003:
2999:
2995:
2991:
2987:
2986:
2985:
2984:
2983:
2981:
2977:
2973:
2969:
2965:
2958:
2945:
2940:
2936:
2930:
2927:
2922:
2917:
2913:
2907:
2904:
2899:
2894:
2890:
2884:
2881:
2876:
2871:
2867:
2861:
2858:
2853:
2848:
2844:
2838:
2835:
2830:
2825:
2821:
2815:
2812:
2802:
2792:
2788:
2784:
2779:
2774:
2770:
2766:
2750:
2733:
2729:
2726:
2725:
2722:
2718:
2714:
2710:
2707:
2706:
2702:
2691:
2687:
2683:
2679:
2675:
2671:
2667:
2666:Pascal matrix
2663:
2659:
2640:
2637:
2634:
2628:
2623:
2609:
2608:
2607:
2603:
2599:
2594:
2593:
2592:
2588:
2584:
2580:
2576:
2573:
2550:
2540:
2534:
2531:
2528:
2518:
2517:
2499:
2483:
2480:
2478:
2474:
2470:
2466:
2462:
2459:
2457:
2453:
2449:
2445:
2442:
2440:
2436:
2432:
2428:
2425:
2424:
2423:
2422:
2418:
2414:
2396:
2392:
2388:
2385:
2382:
2377:
2373:
2369:
2364:
2360:
2343:
2339:
2335:
2331:
2327:
2326:
2325:
2322:
2319:
2313:
2312:
2307:
2303:
2297:
2296:
2292:
2288:
2284:
2277:
2274:
2273:
2272:
2266:
2262:
2258:
2254:
2250:
2249:
2244:
2243:
2242:
2240:
2236:
2232:
2231:80.100.243.19
2228:
2221:
2214:
2212:
2210:
2206:
2202:
2198:
2180:
2176:
2172:
2169:
2149:
2145:
2141:
2138:
2116:
2112:
2091:
2088:
2085:
2073:
2061:
2057:
2053:
2049:
2048:
2047:
2043:
2039:
2021:
2012:
1987:
1983:
1960:
1956:
1933:
1924:
1899:
1890:
1865:
1861:
1852:
1851:
1850:
1846:
1842:
1824:
1820:
1811:
1807:
1806:
1805:
1801:
1797:
1779:
1770:
1745:
1741:
1732:
1731:
1730:
1726:
1722:
1703:
1700:
1697:
1693:
1687:
1684:
1681:
1673:
1670:
1659:
1656:
1653:
1649:
1625:
1616:
1591:
1587:
1578:
1577:
1576:
1573:
1552:
1543:
1533:The variable
1528:
1526:
1525:
1521:
1517:
1513:
1509:
1501:
1499:
1496:
1492:
1488:
1484:
1480:
1473:
1469:
1465:
1461:
1457:
1453:
1449:
1425:
1421:
1417:
1413:
1412:46.242.13.183
1409:
1408:
1407:
1406:
1402:
1398:
1394:
1390:
1386:
1380:
1374:
1370:
1366:
1362:
1361:
1358:
1354:
1350:
1346:
1342:
1335:
1331:
1330:
1329:
1326:
1322:
1318:
1314:
1310:
1295:
1291:
1287:
1283:
1282:81.130.237.47
1278:
1277:
1276:
1275:
1271:
1267:
1259:
1257:
1246:
1242:
1238:
1235:
1232:
1228:
1227:
1226:
1225:
1224:
1223:
1222:
1221:
1217:
1213:
1208:
1207:
1161:
1160:
1159:
1158:
1155:
1151:
1147:
1144:
1127:
1124:
1121:
1117:
1113:
1108:
1105:
1102:
1094:
1091:
1088:
1082:
1077:
1073:
1069:
1064:
1061:
1058:
1054:
1031:
1023:
1020:
1017:
1011:
1006:
1002:
998:
993:
990:
987:
983:
974:
970:
969:
968:
967:
963:
959:
954:
953:
945:
943:
940:
936:
932:
928:
924:
915:
911:
904:
902:
901:
897:
893:
885:
880:
876:
871:
868:
864:
860:
855:
851:
850:
849:
848:
844:
840:
832:
826:
822:
818:
814:
810:
809:
808:
807:
804:
800:
796:
795:Michael Hardy
792:
791:
790:
789:
785:
781:
774:
771:
770:
769:
767:
763:
755:
751:
747:
743:
742:Michael Hardy
735:
731:
729:
707:
699:
696:
693:
685:
682:
679:
674:
670:
666:
661:
657:
653:
648:
644:
640:
635:
631:
625:
622:
619:
614:
611:
608:
604:
596:
595:
594:
593:
592:
591:
590:
587:
585:
578:
574:
570:
566:
564:
547:
541:
538:
533:
521:
518:
513:
508:
504:
492:
484:
481:
478:
475:
472:
468:
462:
458:
446:
442:
439:
436:
422:
417:
414:
411:
407:
400:
397:
394:
390:
385:
380:
376:
370:
365:
362:
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355:
347:
346:
345:
342:
341:
340:
339:
338:
335:
334:
333:
332:
328:
324:
320:
312:
306:
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279:
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261:
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142:
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129:
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108:
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78:
75:
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66:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
3067:
3049:
3046:
3043:
3040:
3037:
3034:
3031:
2962:— Preceding
2959:
2803:
2800:
2783:67.198.37.16
2735:
2727:
2713:37.76.55.242
2708:
2700:
2481:
2460:
2443:
2426:
2350:
2329:
2323:
2317:
2314:
2280:
2270:
2225:— Preceding
2222:
2218:
2195:— Preceding
2077:
1532:
1505:
1477:— Preceding
1429:
1395:
1391:
1387:
1384:
1339:— Preceding
1332:Both are in
1307:— Preceding
1299:
1264:
1253:
1209:
1202:
955:
949:
921:— Preceding
916:
912:
908:
889:
867:Zorn's lemma
862:
839:Eric Kvaalen
836:
812:
777:
772:
761:
759:
733:
576:
575:, not up to
572:
568:
529:rather than
316:
297:Eric Kvaalen
282:Eric Kvaalen
253:
231:Eric Kvaalen
179:special case
178:
176:
161:
137:Low-priority
136:
96:
62:Low‑priority
40:WikiProjects
3051:Kaizen Grey
2709:Don't merge
1438:= (n+1) * F
1381:Proof style
567:(the index
258:—Preceding
112:Mathematics
103:mathematics
59:Mathematics
3086:Categories
2992:note 1. --
2657:necessary.
2448:XOR'easter
2968:HarryToby
2482:Not merge
2328:There is
2002:as "like
1572:Ntropyman
892:Twin Bird
817:Wirkstoff
766:MathWorld
323:MathFacts
164:Crackling
2976:contribs
2964:unsigned
2469:D.Lazard
2334:Felix QW
2253:D.Lazard
2227:unsigned
2197:unsigned
1810:D.Lazard
1721:D.Lazard
1491:contribs
1479:unsigned
1353:contribs
1341:unsigned
1321:contribs
1309:unsigned
1168:n lim (Σ
935:contribs
923:unsigned
272:contribs
260:unsigned
3012:Bio.oid
2994:Bio.oid
2734:, some
2682:Bio.oid
2583:Bio.oid
2461:Support
2444:Support
2427:Support
2052:Bob.v.R
1841:Bob.v.R
1796:Bob.v.R
1516:Jsondow
1508:Jsondow
1472:A253713
1468:A253712
1464:A253711
1460:A253710
1456:A253637
1452:A253636
1448:A250212
1266:Bob.v.R
875:Dominus
139:on the
30:C-class
1975:, and
1046:, and
975:, for
863:called
764:while
36:scale.
2411:. --
2304:into
1853:Both
1733:Both
1444:(m+1)
1436:(n,m)
1397:Haklo
1365:Haklo
1296:Doubt
1237:Haklo
1196:n)/(Σ
1188:n)/(Σ
1180:n)/(Σ
1172:n)/(Σ
1146:Haklo
927:Haklo
833:Move?
780:Maxal
579:+ 1).
499:with
3074:talk
3055:talk
3016:talk
2998:talk
2972:talk
2787:talk
2717:talk
2686:talk
2602:talk
2587:talk
2473:talk
2452:talk
2435:talk
2417:talk
2338:talk
2291:talk
2257:talk
2235:talk
2205:talk
2078:For
2056:talk
2042:talk
1880:and
1845:talk
1800:talk
1760:and
1725:talk
1520:talk
1512:talk
1487:talk
1416:talk
1401:talk
1369:talk
1349:talk
1317:talk
1286:talk
1270:talk
1241:talk
1216:talk
1150:talk
962:talk
931:talk
896:talk
886:NPOV
879:talk
852:The
843:talk
821:talk
799:talk
784:talk
746:talk
327:talk
301:talk
286:talk
268:talk
235:talk
2771:or
2598:JBL
2581:.--
2413:JBL
2285:. —
2193:.
2131:is
1808:If
1606:to
1470:,
1466:,
1462:,
1458:,
1454:,
1450:,
1442:- F
1440:(m)
1432:(m)
1336:.
1304:.
131:Low
3088::
3076:)
3057:)
3018:)
3010:--
3000:)
2978:)
2974:•
2931:20
2923:−
2885:10
2877:−
2826:10
2816:10
2789:)
2719:)
2688:)
2672:,
2668:,
2629:∉
2604:)
2596:--
2589:)
2541:∈
2475:)
2467:.
2454:)
2437:)
2419:)
2386:…
2340:)
2293:)
2259:)
2237:)
2207:)
2181:20
2170:−
2150:12
2139:−
2058:)
2044:)
2016:¯
1928:¯
1894:¯
1847:)
1839:?
1802:)
1774:¯
1727:)
1671:−
1665:→
1620:¯
1547:¯
1522:)
1493:)
1489:•
1418:)
1403:)
1371:)
1355:)
1351:•
1323:)
1319:•
1288:)
1272:)
1243:)
1218:)
1210:--
1152:)
1114:−
1070:−
999:−
964:)
956:--
937:)
933:•
898:)
845:)
823:)
801:)
786:)
748:)
697:−
683:⋯
623:−
605:∑
534:−
482:−
408:∑
356:∑
329:)
303:)
288:)
274:)
270:•
237:)
193:−
169:)
3072:(
3053:(
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2996:(
2970:(
2946:.
2941:2
2937:n
2928:3
2918:4
2914:n
2908:2
2905:1
2900:+
2895:6
2891:n
2882:7
2872:8
2868:n
2862:4
2859:3
2854:+
2849:9
2845:n
2839:2
2836:1
2831:+
2822:n
2813:1
2785:(
2751:2
2746:C
2715:(
2684:(
2644:)
2641:1
2638:,
2635:1
2632:(
2624:2
2619:C
2600:(
2585:(
2551:2
2546:C
2538:)
2535:1
2532:,
2529:1
2526:(
2500:2
2495:C
2471:(
2450:(
2433:(
2415:(
2397:m
2393:n
2389:+
2383:+
2378:m
2374:2
2370:+
2365:m
2361:1
2336:(
2289:(
2255:(
2233:(
2203:(
2177:/
2173:3
2146:/
2142:1
2117:2
2113:n
2092:9
2089:=
2086:p
2054:(
2040:(
2022:m
2013:A
1988:m
1984:A
1961:m
1957:G
1934:m
1925:A
1900:m
1891:A
1866:m
1862:A
1843:(
1825:m
1821:A
1798:(
1780:m
1771:A
1746:m
1742:A
1723:(
1704:j
1701:,
1698:i
1694:a
1688:j
1685:+
1682:i
1678:)
1674:1
1668:(
1660:j
1657:,
1654:i
1650:a
1626:m
1617:A
1592:m
1588:A
1553:m
1544:A
1518:(
1510:(
1485:(
1414:(
1399:(
1367:(
1347:(
1315:(
1302:1
1284:(
1268:(
1239:(
1214:(
1198:n
1194:n
1190:n
1186:n
1182:n
1178:n
1174:n
1170:n
1148:(
1128:1
1125:+
1122:m
1118:n
1109:1
1106:+
1103:m
1099:)
1095:1
1092:+
1089:n
1086:(
1083:=
1078:n
1074:b
1065:1
1062:+
1059:n
1055:b
1032:m
1028:)
1024:1
1021:+
1018:n
1015:(
1012:=
1007:n
1003:a
994:1
991:+
988:n
984:a
960:(
929:(
894:(
881:)
877:(
873:—
841:(
819:(
797:(
782:(
744:(
737:1
734:B
708:n
703:)
700:1
694:m
691:(
686:+
680:+
675:n
671:2
667:+
662:n
658:1
654:+
649:n
645:0
641:=
636:n
632:k
626:1
620:m
615:0
612:=
609:k
577:p
573:p
569:j
548:)
542:2
539:1
522:2
519:1
514:=
509:1
505:B
493:(
485:j
479:1
476:+
473:p
469:n
463:j
459:B
452:)
447:j
443:1
440:+
437:p
431:(
423:p
418:0
415:=
412:j
401:1
398:+
395:p
391:1
386:=
381:p
377:k
371:n
366:1
363:=
360:k
325:(
299:(
284:(
266:(
233:(
204:p
200:)
196:1
190:(
143:.
42::
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