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- Bernoulli_number abstract "In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers with deep connections to number theory. The values of the first few Bernoulli numbers are B0 = 1, B1 = ±1⁄2, B2 = 1⁄6, B3 = 0, B4 = −1⁄30, B5 = 0, B6 = 1⁄42, B7 = 0, B8 = −1⁄30.If the convention B1 = −1⁄2 is used, this sequence is also known as the first Bernoulli numbers (OEIS A027641 / OEIS A027642 in OEIS); with the convention B1 = +1⁄2 is known as the second Bernoulli numbers (OEIS A164555 / OEIS A027642). Except for this one difference, the first and second Bernoulli numbers agree. Since Bn = 0 for all odd n > 1, and many formulas only involve even-index Bernoulli numbers, some authors write Bn instead of B2n.The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in formulas for the sum of powers of the first positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function.The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jakob Bernoulli, after whom they are named, and independently by Japanese mathematician Seki Kōwa. Seki's discovery was posthumously published in 1712 in his work Katsuyo Sampo; Bernoulli's, also posthumously, in his Ars Conjectandi of 1713. Ada Lovelace's note G on the analytical engine from 1842 describes an algorithm for generating Bernoulli numbers with Babbage's machine. As a result, the Bernoulli numbers have the distinction of being the subject of the first computer program.".
- Bernoulli_number wikiPageExternalLink today-we-broke-the-bernoulli-record-from-the-analytical-engine-to-mathematica.
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- Bernoulli_number wikiPageExternalLink text-idx?c=math;idno=00450002.
- Bernoulli_number wikiPageExternalLink s05dumont.html.
- Bernoulli_number wikiPageExternalLink en.literateprograms.org.
- Bernoulli_number wikiPageExternalLink 02_2_GeneralizedBernoulliRecursion.pdf.
- Bernoulli_number wikiPageExternalLink 04_3_SummingOfLikePowers.pdf.
- Bernoulli_number wikiPageExternalLink bernoulli_en.pdf.
- Bernoulli_number wikiPageExternalLink ComputationAndAsymptoticsOfBernoulliNumbers.
- Bernoulli_number wikiPageExternalLink zwsun.
- Bernoulli_number wikiPageExternalLink purl?GDZPPN002147319.
- Bernoulli_number wikiPageExternalLink purl?GDZPPN002158698.
- Bernoulli_number wikiPageExternalLink bernmm.
- Bernoulli_number wikiPageExternalLink www.bernoulli.org.
- Bernoulli_number wikiPageExternalLink www.bernoulli.org.
- Bernoulli_number wikiPageExternalLink v11i2r19.html.
- Bernoulli_number wikiPageExternalLink vol3.html.
- Bernoulli_number wikiPageExternalLink GDZPPN002146738.
- Bernoulli_number wikiPageExternalLink Paper30-Acta27-2011.pdf.
- Bernoulli_number wikiPageExternalLink sketch.html.
- Bernoulli_number wikiPageExternalLink bernincl.html.
- Bernoulli_number wikiPageExternalLink irregular.html.
- Bernoulli_number wikiPageExternalLink bernoulli.html.
- Bernoulli_number wikiPageID "4964".
- Bernoulli_number wikiPageRevisionID "605430828".
- Bernoulli_number hasPhotoCollection Bernoulli_number.
- Bernoulli_number id "p/b015640".
- Bernoulli_number title "Bernoulli Number".
- Bernoulli_number title "Bernoulli numbers".
- Bernoulli_number urlname "BernoulliNumber".
- Bernoulli_number subject Category:Integer_sequences.
- Bernoulli_number subject Category:Number_theory.
- Bernoulli_number subject Category:Topology.
- Bernoulli_number type Abstraction100002137.
- Bernoulli_number type Arrangement107938773.
- Bernoulli_number type Group100031264.
- Bernoulli_number type IntegerSequences.
- Bernoulli_number type Ordering108456993.
- Bernoulli_number type Sequence108459252.
- Bernoulli_number type Series108457976.
- Bernoulli_number comment "In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers with deep connections to number theory. The values of the first few Bernoulli numbers are B0 = 1, B1 = ±1⁄2, B2 = 1⁄6, B3 = 0, B4 = −1⁄30, B5 = 0, B6 = 1⁄42, B7 = 0, B8 = −1⁄30.If the convention B1 = −1⁄2 is used, this sequence is also known as the first Bernoulli numbers (OEIS A027641 / OEIS A027642 in OEIS); with the convention B1 = +1⁄2 is known as the second Bernoulli numbers (OEIS A164555 / OEIS A027642).".
- Bernoulli_number label "Bernoulli number".
- Bernoulli_number label "Bernoulli-Zahl".
- Bernoulli_number label "Bernoulligetal".
- Bernoulli_number label "Liczby Bernoulliego".
- Bernoulli_number label "Nombre de Bernoulli".
- Bernoulli_number label "Numeri di Bernoulli".
- Bernoulli_number label "Número de Bernoulli".
- Bernoulli_number label "Números de Bernoulli".
- Bernoulli_number label "Числа Бернулли".
- Bernoulli_number label "عدد بيرنولي".
- Bernoulli_number label "ベルヌーイ数".
- Bernoulli_number label "伯努利数".
- Bernoulli_number sameAs Bernoulli-Zahl.
- Bernoulli_number sameAs Número_de_Bernoulli.
- Bernoulli_number sameAs Nombre_de_Bernoulli.
- Bernoulli_number sameAs Numeri_di_Bernoulli.
- Bernoulli_number sameAs ベルヌーイ数.
- Bernoulli_number sameAs 베르누이_수.
- Bernoulli_number sameAs Bernoulligetal.
- Bernoulli_number sameAs Liczby_Bernoulliego.
- Bernoulli_number sameAs Números_de_Bernoulli.
- Bernoulli_number sameAs m.01klw.
- Bernoulli_number sameAs Q694114.
- Bernoulli_number sameAs Q694114.
- Bernoulli_number sameAs Bernoulli_number.
- Bernoulli_number wasDerivedFrom Bernoulli_number?oldid=605430828.
- Bernoulli_number isPrimaryTopicOf Bernoulli_number.