Binomial coefficient (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Binomial coefficient" in English language version.

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ams.org (Global: 451^st place; English: 277^th place)

mathscinet.ams.org

See (Graham, Knuth & Patashnik 1994), which also defines ${\tbinom {n}{k}}=0$ for $k<0$ . Alternative generalizations, such as to two real or complex valued arguments using the Gamma function assign nonzero values to ${\tbinom {n}{k}}$ for $k<0$ , but this causes most binomial coefficient identities to fail, and thus is not widely used by the majority of definitions. One such choice of nonzero values leads to the aesthetically pleasing "Pascal windmill" in Hilton, Holton and Pedersen, Mathematical reflections: in a room with many mirrors, Springer, 1997, but causes even Pascal's identity to fail (at the origin). Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (February 1994). Concrete Mathematics - A foundation for computer science (2nd ed.). Reading, MA, USA: Addison-Wesley Professional. pp. 154–155. ISBN 0-201-55802-5. MR 1397498.
Boardman, Michael (2004), "The Egg-Drop Numbers", Mathematics Magazine, 77 (5): 368–372, doi:10.2307/3219201, JSTOR 3219201, MR 1573776, it is well known that there is no closed form (that is, direct formula) for the partial sum of binomial coefficients.
Munarini, Emanuele (2011), "Riordan matrices and sums of harmonic numbers" (PDF), Applicable Analysis and Discrete Mathematics, 5 (2): 176–200, doi:10.2298/AADM110609014M, MR 2867317.

archive.org (Global: 6^th place; English: 6^th place)

Higham (1998) Higham, Nicholas J. (1998). Handbook of writing for the mathematical sciences. SIAM. p. 25. ISBN 0-89871-420-6.
Uspensky 1937, p. 18 Uspensky, James (1937), Introduction to Mathematical Probability, McGraw-Hill

arxiv.org (Global: 69^th place; English: 59^th place)

Ruiz, Sebastian (1996). "An Algebraic Identity Leading to Wilson's Theorem". The Mathematical Gazette. 80 (489): 579–582. arXiv:math/0406086. doi:10.2307/3618534. JSTOR 3618534. S2CID 125556648.
Farhi, Bakir (2007). "Nontrivial lower bounds for the least common multiple of some finite sequence of integers". Journal of Number Theory. 125 (2): 393–411. arXiv:0803.0290. doi:10.1016/j.jnt.2006.10.017. S2CID 115167580.

books.google.com (Global: 3^rd place; English: 3^rd place)

Muir, Thomas (1902). "Note on Selected Combinations". Proceedings of the Royal Society of Edinburgh.

doi.org (Global: 2^nd place; English: 2^nd place)

Boardman, Michael (2004), "The Egg-Drop Numbers", Mathematics Magazine, 77 (5): 368–372, doi:10.2307/3219201, JSTOR 3219201, MR 1573776, it is well known that there is no closed form (that is, direct formula) for the partial sum of binomial coefficients.
see induction developed in eq (7) p. 1389 in Aupetit, Michael (2009), "Nearly homogeneous multi-partitioning with a deterministic generator", Neurocomputing, 72 (7–9): 1379–1389, doi:10.1016/j.neucom.2008.12.024, ISSN 0925-2312.
Ruiz, Sebastian (1996). "An Algebraic Identity Leading to Wilson's Theorem". The Mathematical Gazette. 80 (489): 579–582. arXiv:math/0406086. doi:10.2307/3618534. JSTOR 3618534. S2CID 125556648.
Farhi, Bakir (2007). "Nontrivial lower bounds for the least common multiple of some finite sequence of integers". Journal of Number Theory. 125 (2): 393–411. arXiv:0803.0290. doi:10.1016/j.jnt.2006.10.017. S2CID 115167580.
Munarini, Emanuele (2011), "Riordan matrices and sums of harmonic numbers" (PDF), Applicable Analysis and Discrete Mathematics, 5 (2): 176–200, doi:10.2298/AADM110609014M, MR 2867317.

jstor.org (Global: 26^th place; English: 20^th place)

Boardman, Michael (2004), "The Egg-Drop Numbers", Mathematics Magazine, 77 (5): 368–372, doi:10.2307/3219201, JSTOR 3219201, MR 1573776, it is well known that there is no closed form (that is, direct formula) for the partial sum of binomial coefficients.
Ruiz, Sebastian (1996). "An Algebraic Identity Leading to Wilson's Theorem". The Mathematical Gazette. 80 (489): 579–582. arXiv:math/0406086. doi:10.2307/3618534. JSTOR 3618534. S2CID 125556648.

nb.rs (Global: 3,800^th place; English: 8,978^th place)

doiserbia.nb.rs

Munarini, Emanuele (2011), "Riordan matrices and sums of harmonic numbers" (PDF), Applicable Analysis and Discrete Mathematics, 5 (2): 176–200, doi:10.2298/AADM110609014M, MR 2867317.

semanticscholar.org (Global: 11^th place; English: 8^th place)

api.semanticscholar.org

Ruiz, Sebastian (1996). "An Algebraic Identity Leading to Wilson's Theorem". The Mathematical Gazette. 80 (489): 579–582. arXiv:math/0406086. doi:10.2307/3618534. JSTOR 3618534. S2CID 125556648.
Farhi, Bakir (2007). "Nontrivial lower bounds for the least common multiple of some finite sequence of integers". Journal of Number Theory. 125 (2): 393–411. arXiv:0803.0290. doi:10.1016/j.jnt.2006.10.017. S2CID 115167580.

springer.com (Global: 274^th place; English: 309^th place)

see e.g. Ash (1990, p. 121) or Flum & Grohe (2006, p. 427). Ash, Robert B. (1990) [1965]. Information Theory. Dover Publications, Inc. ISBN 0-486-66521-6. Flum, Jörg; Grohe, Martin (2006). Parameterized Complexity Theory. Springer. ISBN 978-3-540-29952-3. Archived from the original on 2007-11-18. Retrieved 2017-08-28.

web.archive.org (Global: 1^st place; English: 1^st place)

see e.g. Ash (1990, p. 121) or Flum & Grohe (2006, p. 427). Ash, Robert B. (1990) [1965]. Information Theory. Dover Publications, Inc. ISBN 0-486-66521-6. Flum, Jörg; Grohe, Martin (2006). Parameterized Complexity Theory. Springer. ISBN 978-3-540-29952-3. Archived from the original on 2007-11-18. Retrieved 2017-08-28.

worldcat.org (Global: 5^th place; English: 5^th place)

search.worldcat.org

see induction developed in eq (7) p. 1389 in Aupetit, Michael (2009), "Nearly homogeneous multi-partitioning with a deterministic generator", Neurocomputing, 72 (7–9): 1379–1389, doi:10.1016/j.neucom.2008.12.024, ISSN 0925-2312.
Spencer, Joel; Florescu, Laura (2014). Asymptopia. Student mathematical library. Vol. 71. AMS. p. 66. ISBN 978-1-4704-0904-3. OCLC 865574788.
Spencer, Joel; Florescu, Laura (2014). Asymptopia. Student mathematical library. Vol. 71. AMS. p. 59. ISBN 978-1-4704-0904-3. OCLC 865574788.

Binomial coefficient (English Wikipedia)

ams.org (Global: 451st place; English: 277th place)

mathscinet.ams.org

archive.org (Global: 6th place; English: 6th place)

arxiv.org (Global: 69th place; English: 59th place)

books.google.com (Global: 3rd place; English: 3rd place)

doi.org (Global: 2nd place; English: 2nd place)

jstor.org (Global: 26th place; English: 20th place)

nb.rs (Global: 3,800th place; English: 8,978th place)