二項係数
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出典^ Lilavati Section 6, Chapter 4 (see Knuth (1997)).
^ Higham (1998)
^ Shilov (1977, p. 92)
^ Holton, Pedersen (1997), Mathematical Reflections: In a Room With Many Mirrors, Springer, .mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation.cs-ja1 q,.mw-parser-output .citation.cs-ja2 q{quotes:"「""」""『""』"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:#d33}.mw-parser-output .cs1-visible-error{color:#d33}.mw-parser-output .cs1-maint{display:none;color:#3a3;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}ISBN 978-1-4612-1932-3
^ Thomas, Muir (1904). “Note on selected combinations”. Proceedings of the Royal Society of Edinburgh. doi:10.1017/S0370164600007768. https://books.google.co.jp/books/reader?id=EN8vAAAAIAAJ&output=reader&pg=GBS.PA102&redir_esc=y&hl=ja.
^ Boardman, Michael (2004), “The Egg-drop numbers”, Mathematics Magazine 77 (5): 368-372, JSTOR 3219201, MR1573776, https://jstor.org/stable/3219201, "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. doi:10.2307/3618534. ⇒http://www.jstor.org/stable/3618534.
^ Knuth 1997, p. 30.
^ see e.g. Ash (1990, p. 121) or Flum & Grohe (2006, p. 427).
^ Munarini, Emanuele (2011), “Riordan matrices and sums of harmonic numbers”, Applicable Analysis and Discrete Mathematics 5 (2): 176-200, doi:10.2298/AADM110609014M, MR2867317 .
参考文献.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-100{font-size:100%}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}
Ash, Robert B. (1990) [1965]. Information Theory. Dover Publications, Inc.. ISBN 0-486-66521-6. http://www.amazon.com/Information-Theory-Dover-Books-Mathematics/dp/0486665216
Benjamin, Arthur T.; Jennifer, Quinn (2003). Proofs that Really Count: The Art of Combinatorial Proof. Mathematical Association of America. ISBN 978-0-88385-333-7. https://www.maa.org/press/books/proofs-that-really-count-the-art-of-combinatorial-proof
Bryant, Victor (1993). Aspects of Combinatorics. Cambridge University Press. ISBN 0-521-41974-3
Flum, Jorg; Grohe, Martin (2006). ⇒Parameterized Complexity Theory. Springer. ISBN 978-3-540-29952-3. ⇒http://www.springer.com/east/home/generic/search/results?SGWID=5-40109-22-141358322-0
Fowler, David (1996-01). “The binomial coefficient function”. The American Mathematical Monthly (Mathematical Association of America) 103 (1): 1-17. doi:10.2307/2975209. JSTOR 2975209
Goetgheluck, P. (1987). “Computing binomial coefficients”. American Math. Monthly 94: 360-365. doi:10.2307/2323099.
Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1994). Concrete Mathematics (Second ed.). Addison-Wesley. pp. 153-256. ISBN 0-201-55802-5
Higham, Nicholas J. (1998). Handbook of Writing for the Mathematical Sciences. SIAM. p. 25. ISBN 0-89871-420-6
Knuth, Donald E. (1997). The Art of Computer Programming, Volume 1: Fundamental Algorithms (Third ed.). Addison-Wesley. pp. 52-74. ISBN 0-201-89683-4
Singmaster, David (1974). “Notes on binomial coefficients. III. Any integer divides almost all binomial coefficients”. Journal of the London Mathematical Society 8 (3): 555-560. doi:10.1112/jlms/s2-8.3.555.
Shilov, G. E. (1977). Linear Algebra. Dover Publications. ISBN 978-0-486-63518-7
外部リンク
『二項係数の有名公式一覧と2つの証明方針』 - 高校数学の美しい物語
『二項係数の和，二乗和，三乗和』 - 高校数学の美しい物語
Hazewinkel, Michiel, ed. (2001), “Binomial coefficients”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Binomial_coefficients
Andrew Granville (1997). ⇒“Arithmetic properties of binomial coefficients I. Binomial coefficients modulo prime powers”. CMS Conf. Proc 20: 151-162. ⇒http://www.cecm.sfu.ca/organics/papers/granville/Binomial/toppage.html.
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