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== References == {{reflist|refs= <ref name=Weisstein2016>{{mathworld|id=BernoulliNumber |title=Bernoulli Number|mode=cs2}}</ref> <ref name=Selin1997_891>{{citation |editor-last=Selin |editor-first=Helaine | editor-link=Helaine Selin |date=1997 |title=Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures |isbn=0-7923-4066-3 |at=p. 819 (p. 891) |publisher=Springer |bibcode=2008ehst.book.....S}}</ref> <ref name=SmithMikami1914_108>{{citation |last1=Smith |first1=David Eugene |last2=Mikami |first2=Yoshio |date=1914 |title=A history of Japanese mathematics |publisher=Open Court publishing company |page=108 |isbn=9780486434827 |url=https://books.google.com/books?id=pTcQsvfbSu4C}}</ref> <ref name=Kitagawa>{{citation|last=Kitagawa|first=Tomoko L.|date=2021-07-23|title=The Origin of the Bernoulli Numbers: Mathematics in Basel and Edo in the Early Eighteenth Century|journal=The Mathematical Intelligencer|volume=44 |pages=46–56 |language=en|doi=10.1007/s00283-021-10072-y|issn=0343-6993|doi-access=free}}</ref> <ref name=Menabrea1842_noteG>{{citation |last=Menabrea |first=L.F. |date=1842 |title=Sketch of the Analytic Engine invented by Charles Babbage, with notes upon the Memoir by the Translator Ada Augusta, Countess of Lovelace |journal=Bibliothèque Universelle de Genève |volume=82 |at=See ''Note G'' |url=http://www.fourmilab.ch/babbage/sketch.html}}</ref> <ref name=Jacobi1834>{{citation |last=Jacobi |first=C.G.J. |author-link=Carl Gustav Jacob Jacobi |title=De usu legitimo formulae summatoriae Maclaurinianae |journal=Journal für die reine und angewandte Mathematik |volume=12 |date=1834 |pages=263–272 |url=https://zenodo.org/record/1448824 }}</ref> <ref name="Saalschütz1893">{{citation |last=Saalschütz |first=Louis |date=1893 |title=Vorlesungen über die Bernoullischen Zahlen, ihren Zusammenhang mit den Secanten-Coefficienten und ihre wichtigeren Anwendungen |place=Berlin |publisher=Julius Springer |isbn=978-3-662-40711-0 |url=https://books.google.com/books?id=-H8olxg5f1gC }}.</ref> <ref name=BuhlerCraErnMetShokrollahi2001>{{citation |last1=Buhler |first1=J. |last2=Crandall |first2=R. |last3=Ernvall |first3=R. |last4=Metsankyla |first4=T. |last5=Shokrollahi |first5=M. |title=Irregular Primes and Cyclotomic Invariants to 12 Million |journal=Journal of Symbolic Computation |volume=31 |issue=1–2 |date=2001 |pages=89–96 |doi=10.1006/jsco.1999.1011|doi-access=free }}</ref> <ref name=Harvey2010>{{citation |last=Harvey |first=David |date=2010 |title=A multimodular algorithm for computing Bernoulli numbers |journal=Math. Comput. |volume=79 |issue=272 |pages=2361–2370 |arxiv=0807.1347 |doi=10.1090/S0025-5718-2010-02367-1 |zbl=1215.11016 |s2cid=11329343 }}</ref> <ref name=Kellner2002>{{citation |last=Kellner |first=Bernd |date=2002 |title=Program Calcbn – A program for calculating Bernoulli numbers |url=http://www.bernoulli.org/}}.</ref> <ref name=Pavlyk29Apr2008>{{citation |last=Pavlyk |first=Oleksandr |date=29 April 2008 |title=Today We Broke the Bernoulli Record: From the Analytical Engine to Mathematica |website=Wolfram News |url=http://blog.wolfram.com/2008/04/29/today-we-broke-the-bernoulli-record-from-the-analytical-engine-to-mathematica/}}.</ref> <ref name=GuoZeng2005>{{citation |last1=Guo |first1=Victor J. W. |last2=Zeng |first2=Jiang |date=30 August 2005 |title=A q-Analogue of Faulhaber's Formula for Sums of Powers |journal=The Electronic Journal of Combinatorics |volume=11 |issue=2 |doi=10.37236/1876 |bibcode=2005math......1441G |arxiv=math/0501441 |s2cid=10467873 }}</ref> <!--<ref name=Comtet1974>{{citation |last=Comtet |first=L. |date=1974 |title=Advanced combinatorics. The art of finite and infinite expansions|edition=Revised and Enlarged |location=Dordrecht-Boston |publisher=D. Reidel Publ.}}</ref> --> <ref name=Rademacher1973>{{citation |last=Rademacher |first=H. |date=1973 |title=Analytic Number Theory |location=New York City |publisher=Springer-Verlag}}.</ref> <ref name=Boole1880>{{citation |last=Boole |first=G. |date=1880 |title=A treatise of the calculus of finite differences |edition=3rd |place=London |publisher=Macmillan}}.</ref> <ref name=Gould1972>{{citation |last=Gould |first=Henry W. |title=Explicit formulas for Bernoulli numbers |journal=Amer. Math. Monthly |volume=79 |issue=1 |date=1972 |pages=44–51 |doi=10.2307/2978125 |jstor=2978125}}</ref> <ref name=Apostol2010_197>{{citation |last=Apostol |first=Tom M. |date=2010 |title=Introduction to Analytic Number Theory |publisher=Springer-Verlag |page=197}}</ref> <ref name=Woon1997>{{citation |last=Woon |first=S. C. |title=A tree for generating Bernoulli numbers |jstor=2691054 |journal=Math. Mag. |volume=70 |issue=1 |date=1997 |pages=51–56 |doi=10.2307/2691054}}</ref> <ref name=Elkies2003>{{citation |last=Elkies |first=N. D. |author-link=Noam Elkies |title=On the sums Sum_(k=-infinity...infinity) (4k+1)^(-n) |journal=Amer. Math. Monthly |volume=110 |issue=7 |date=2003 |pages=561–573 |doi=10.2307/3647742 |arxiv=math.CA/0101168 |jstor=3647742 }}</ref> <ref name=Euler1735>{{citation |last=Euler |first=Leonhard |author-link=Leonhard Euler |title=De summis serierum reciprocarum |journal=Opera Omnia |volume=I.14, E 41 |date=1735 |pages=73–86 |arxiv=math/0506415 |bibcode=2005math......6415E }}</ref> <ref name=Seidel1877>{{citation |last=Seidel |first=L. |title=Über eine einfache Entstehungsweise der Bernoullischen Zahlen und einiger verwandten Reihen |journal=Sitzungsber. Münch. Akad. |volume=4 |date=1877 |pages=157–187}}</ref> <ref name=Dumont1981>{{citation |last=Dumont |first=D. |date=1981 |title=Matrices d'Euler-Seidel |journal=Séminaire Lotharingien de Combinatoire |volume=B05c |url=https://www.mat.univie.ac.at/~slc/opapers/s05dumont.html}}</ref> <ref name=KnuthBuckholtz1967>{{citation |last1=Knuth |first1=D. E. |author-link1= Donald Knuth |last2=Buckholtz |first2=T. J. |date=1967 |title=Computation of Tangent, Euler, and Bernoulli Numbers |journal=Mathematics of Computation |volume=21 |issue=100 |pages=663–688 |publisher=American Mathematical Society |doi=10.2307/2005010 |doi-access=free |jstor=2005010 }}</ref> <ref name=Arnold1991>{{citation |last=Arnold |first=V. I. |date=1991 |title=Bernoulli-Euler updown numbers associated with function singularities, their combinatorics and arithmetics |journal=Duke Math. J. |volume=63 |issue=2 |pages=537–555 |doi=10.1215/s0012-7094-91-06323-4}}</ref> <ref name="André1879">{{citation |last=André |first=D. |title=Développements de sec x et tan x |journal=C. R. Acad. Sci. |volume=88 |date=1879 |pages=965–967}}</ref> <ref name="André1881">{{citation |last=André |first=D. |title=Mémoire sur les permutations alternées |journal=Journal de Mathématiques Pures et Appliquées |volume=7 |date=1881 |pages=167–184}}</ref> <ref name=Kummer1850>{{citation |last=Kummer |first=E. E. |author-link=Ernst Kummer |title=Allgemeiner Beweis des Fermat'schen Satzes, dass die Gleichung x<sup>λ</sup> + y<sup>λ</sup> = z<sup>λ</sup> durch ganze Zahlen unlösbar ist, für alle diejenigen Potenz-Exponenten λ, welche ungerade Primzahlen sind und in den Zählern der ersten (λ-3)/2 Bernoulli'schen Zahlen als Factoren nicht vorkommen |journal=J. Reine Angew. Math. |volume=40 |date=1850 |pages=131–138 |url=http://www.digizeitschriften.de/resolveppn/GDZPPN002146738}}</ref> <ref name=Kummer1851>{{citation |last=Kummer |first=E. E. |author-link=Ernst Kummer |title=Über eine allgemeine Eigenschaft der rationalen Entwicklungscoefficienten einer bestimmten Gattung analytischer Functionen |journal=J. Reine Angew. Math. |volume=1851 |issue=41 |date=1851 |pages=368–372 |url=http://eudml.org/doc/147490 }}</ref> <ref name=vonStaudt1840>{{citation |last=von Staudt |first=K. G. Ch. |title=Beweis eines Lehrsatzes, die Bernoullischen Zahlen betreffend |journal=Journal für die reine und angewandte Mathematik |volume=21 |date=1840 |pages=372–374}}</ref> <ref name=Clausen1840>{{citation |last=Clausen |first=Thomas |title=Lehrsatz aus einer Abhandlung über die Bernoullischen Zahlen |journal=Astron. Nachr. |volume=17 |date=1840 |issue=22 |pages=351–352 |doi=10.1002/asna.18400172205}}</ref> <ref name=Riesz1916>{{citation |last=Riesz |first=M. |title=Sur l'hypothèse de Riemann |journal=Acta Mathematica |volume=40 |date=1916 |pages=185–90 |doi=10.1007/BF02418544 |doi-access=free }}</ref> <ref name=Neukirch1999_VII2>{{Neukirch ANT|mode=cs2}} §VII.2.</ref> <ref name=Malenfant2011>{{cite arXiv |eprint=1103.1585 |last=Malenfant |first=Jerome |date=2011 |title=Finite, closed-form expressions for the partition function and for Euler, Bernoulli, and Stirling numbers |class=math.NT|mode=cs2}}</ref> <ref name=vonEttingshausen1827>{{citation |last=von Ettingshausen |first=A. |date=1827 |title=Vorlesungen über die höhere Mathematik |volume=1 |location=Vienna |publisher=Carl Gerold}}</ref> <ref name=Carlitz1968>{{citation |last=Carlitz |first=L. |title=Bernoulli Numbers |journal=[[Fibonacci Quarterly]] |volume=6 |pages=71–85 |date=1968|issue=3 |doi=10.1080/00150517.1968.12431229 }}</ref> <ref name=AgohDilcher2008>{{citation |last1=Agoh |first1=Takashi|last2=Dilcher |first2=Karl |title=Reciprocity Relations for Bernoulli Numbers |journal=American Mathematical Monthly |volume=115 |issue=3 |date=2008 |pages=237–244 |jstor=27642447 |doi=10.1080/00029890.2008.11920520 |s2cid=43614118 }}</ref> <ref name=Charollois-Sczech>{{citation |last1=Charollois |first1=Pierre |last2=Sczech |first2=Robert |year=2016 |title=Elliptic Functions According to Eisenstein and Kronecker: An Update |journal=EMS Newsletter |language=en |volume=2016-9 |issue=101 |pages=8–14 |doi=10.4171/NEWS/101/4 |s2cid=54504376 |issn=1027-488X|doi-access=free }}</ref> <ref name=BK>{{citation |last1=Bannai |first1=Kenichi |last2=Kobayashi |first2=Shinichi |year=2010 |title=Algebraic theta functions and the p-adic interpolation of Eisenstein-Kronecker numbers |url=https://projecteuclid.org/journals/duke-mathematical-journal/volume-153/issue-2/Algebraic-theta-functions-and-the-p-adic-interpolation-of-Eisenstein/10.1215/00127094-2010-024.full |journal=[[Duke Mathematical Journal]] |volume=153 |issue=2 |doi=10.1215/00127094-2010-024 |arxiv=math/0610163 |s2cid=9262012 |issn=0012-7094}}</ref> }}
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