Pascal's triangle (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Pascal's triangle" in English language version.

Last modified: 11:46:05, 21 August 2026 (UTC)

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adsabs.harvard.edu (Global: 14^th place; English: 9^th place)

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Selin, Helaine (2008-03-12). Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. Springer Science & Business Media. p. 132. Bibcode:2008ehst.book.....S. ISBN 9781402045592. Other, lost works of al-Karaji's are known to have dealt with inderterminate algebra, arithmetic, inheritance algebra, and the construction of buildings. Another contained the first known explanation of the arithmetical (Pascal's) triangle; the passage in question survived through al-Sama'wal's Bahir (twelfth century) which heavily drew from the Badi.
For a similar example, see e.g. Hore, P. J. (1983), "Solvent suppression in Fourier transform nuclear magnetic resonance", Journal of Magnetic Resonance, 55 (2): 283–300, Bibcode:1983JMagR..55..283H, doi:10.1016/0022-2364(83)90240-8.

ams.org (Global: 434^th place; English: 249^th place)

mathscinet.ams.org

Coolidge, J. L. (1949), "The story of the binomial theorem", The American Mathematical Monthly, 56 (3): 147–157, doi:10.2307/2305028, JSTOR 2305028, MR 0028222.
Fine, N. J. (1947), "Binomial coefficients modulo a prime", American Mathematical Monthly, 54 (10): 589–592, doi:10.2307/2304500, JSTOR 2304500, MR 0023257. See in particular Theorem 2, which gives a generalization of this fact for all prime moduli.
Hinz, Andreas M. (1992), "Pascal's triangle and the Tower of Hanoi", The American Mathematical Monthly, 99 (6): 538–544, doi:10.2307/2324061, JSTOR 2324061, MR 1166003. Hinz attributes this observation to an 1891 book by Édouard Lucas, Théorie des nombres (p. 420).

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

Alsdorf, Ludwig (1991) [1933]. "The Pratyayas: Indian Contribution to Combinatorics" (PDF). Indian Journal of History of Science. 26 (1): 17–61. Translated by S. R. Sarma from " $π$ Die Pratyayas. Ein Beitrag zur indischen Mathematik". Zeitschrift für Indologie und Iranistik. 9: 97–157. 1933.
Bag, Amulya Kumar (1966). "Binomial theorem in ancient India" (PDF). Indian Journal of History of Science. 1 (1): 68–74.
Tertiary sources:
Sen, Samarendra Nath (1971). "Mathematics". In Bose, D. M. (ed.). A Concise History Of Science In India. Indian National Science Academy. Ch. 3, pp. 136–212, esp. "Permutations, Combinations and Pascal Triangle", pp. 156–157.
Fowler, David H. (1996). "The Binomial Coefficient Function". The American Mathematical Monthly. 103 (1): 1–17, esp. §4 "A Historical Note", pp. 10–17. doi:10.2307/2975209. JSTOR 2975209.

arxiv.org (Global: 49^th place; English: 31^st place)

Teimoori Faal, Hossein; Khodakarami, Hasan (2023). "Khayyam-Pascal Determinantal Arrays, Star of David Rule and Log-Concavity". arXiv:2302.01637 [math.CO].

bnf.fr (Global: 61^st place; English: 357^th place)

gallica.bnf.fr

Pascal, Blaise (1665). Traité du triangle arithmétique, avec quelques autres petits traitez sur la mesme matière. Par Monsieur Pascal (in French).

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

Peter Fox (1998). Cambridge University Library: the great collections. Cambridge University Press. p. 13. ISBN 978-0-521-62647-7.
Selin, Helaine (2008-03-12). Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. Springer Science & Business Media. p. 132. Bibcode:2008ehst.book.....S. ISBN 9781402045592. Other, lost works of al-Karaji's are known to have dealt with inderterminate algebra, arithmetic, inheritance algebra, and the construction of buildings. Another contained the first known explanation of the arithmetical (Pascal's) triangle; the passage in question survived through al-Sama'wal's Bahir (twelfth century) which heavily drew from the Badi.
Rashed, R. (1994-06-30). The Development of Arabic Mathematics: Between Arithmetic and Algebra. Springer Science & Business Media. p. 63. ISBN 978-0-7923-2565-9.
Sidoli, Nathan; Brummelen, Glen Van (2013-10-30). From Alexandria, Through Baghdad: Surveys and Studies in the Ancient Greek and Medieval Islamic Mathematical Sciences in Honor of J.L. Berggren. Springer Science & Business Media. p. 54. ISBN 9783642367366. However, the use of binomial coefficients by Islamic mathematicians of the eleventh century, in a context which had deep roots in Islamic mathematics, suggests strongly the table was a local discovery - most probably of al-Karaji."
Smith, Karl J. (2010), Nature of Mathematics, Cengage Learning, p. 10, ISBN 9780538737586.
Coxeter, Harold Scott Macdonald (1973-01-01). "Chapter VII: ordinary polytopes in higher space, 7.2: Pyramids, dipyramids and prisms". Regular Polytopes (3rd ed.). Courier Corporation. pp. 118–144. ISBN 978-0-486-61480-9.
Karl, John H. (2012), An Introduction to Digital Signal Processing, Elsevier, p. 110, ISBN 9780323139595.

centre-mersenne.org (Global: low place; English: low place)

ambp.centre-mersenne.org

Kallós, Gábor (2006), "A generalization of Pascal's triangle using powers of base numbers" (PDF), Annales Mathématiques Blaise Pascal, 13 (1): 1–15, doi:10.5802/ambp.211.

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

Coolidge, J. L. (1949), "The story of the binomial theorem", The American Mathematical Monthly, 56 (3): 147–157, doi:10.2307/2305028, JSTOR 2305028, MR 0028222.
Alsdorf, Ludwig (1991) [1933]. "The Pratyayas: Indian Contribution to Combinatorics" (PDF). Indian Journal of History of Science. 26 (1): 17–61. Translated by S. R. Sarma from " $π$ Die Pratyayas. Ein Beitrag zur indischen Mathematik". Zeitschrift für Indologie und Iranistik. 9: 97–157. 1933.
Bag, Amulya Kumar (1966). "Binomial theorem in ancient India" (PDF). Indian Journal of History of Science. 1 (1): 68–74.
Tertiary sources:
Sen, Samarendra Nath (1971). "Mathematics". In Bose, D. M. (ed.). A Concise History Of Science In India. Indian National Science Academy. Ch. 3, pp. 136–212, esp. "Permutations, Combinations and Pascal Triangle", pp. 156–157.
Fowler, David H. (1996). "The Binomial Coefficient Function". The American Mathematical Monthly. 103 (1): 1–17, esp. §4 "A Historical Note", pp. 10–17. doi:10.2307/2975209. JSTOR 2975209.
Khatami, S. M. A. (2020). "Beauties of Khayyam-Pascal triangle". Mathematics and Society. 5 (2): 75–92. doi:10.22108/msci.2021.126885.1413.
Hughes, Barnabas (1 August 1989). "The arithmetical triangle of Jordanus de Nemore". Historia Mathematica. 16 (3): 213–223. doi:10.1016/0315-0860(89)90018-9.
Fowler, David (January 1996). "The Binomial Coefficient Function". The American Mathematical Monthly. 103 (1): 1–17. doi:10.2307/2975209. JSTOR 2975209. See in particular p. 11.
Brothers, H. J. (2012), "Finding e in Pascal's triangle", Mathematics Magazine, 85 (1): 51, doi:10.4169/math.mag.85.1.51, S2CID 218541210.
Brothers, H. J. (2012), "Pascal's triangle: The hidden stor-e", The Mathematical Gazette, 96 (535): 145–148, doi:10.1017/S0025557200004204, S2CID 233356674.
Foster, T. (2014), "Nilakantha's Footprints in Pascal's Triangle", Mathematics Teacher, 108: 247, doi:10.5951/mathteacher.108.4.0246
Fine, N. J. (1947), "Binomial coefficients modulo a prime", American Mathematical Monthly, 54 (10): 589–592, doi:10.2307/2304500, JSTOR 2304500, MR 0023257. See in particular Theorem 2, which gives a generalization of this fact for all prime moduli.
Hinz, Andreas M. (1992), "Pascal's triangle and the Tower of Hanoi", The American Mathematical Monthly, 99 (6): 538–544, doi:10.2307/2324061, JSTOR 2324061, MR 1166003. Hinz attributes this observation to an 1891 book by Édouard Lucas, Théorie des nombres (p. 420).
For a similar example, see e.g. Hore, P. J. (1983), "Solvent suppression in Fourier transform nuclear magnetic resonance", Journal of Magnetic Resonance, 55 (2): 283–300, Bibcode:1983JMagR..55..283H, doi:10.1016/0022-2364(83)90240-8.
Hilton, P.; et al. (1989). "Extending the binomial coefficients to preserve symmetry and pattern". Symmetry 2. In International Series in Modern Applied Mathematics and Computer Science. Pergamon. pp. 89–102. doi:10.1016/B978-0-08-037237-2.50013-1. ISBN 9780080372372..
Hilton, P.; et al. (1989). "Extending the binomial coefficients to preserve symmetry and pattern". Symmetry 2. In International Series in Modern Applied Mathematics and Computer Science. Pergamon. pp. 100–102. doi:10.1016/B978-0-08-037237-2.50013-1. ISBN 9780080372372..
Morton, Robert L. (1964), "Pascal's Triangle and powers of 11", The Mathematics Teacher, 57 (6): 392–394, doi:10.5951/MT.57.6.0392, JSTOR 27957091.
Kallós, Gábor (2006), "A generalization of Pascal's triangle using powers of base numbers" (PDF), Annales Mathématiques Blaise Pascal, 13 (1): 1–15, doi:10.5802/ambp.211.
Hilton, P.; et al. (1989). "Extending the binomial coefficients to preserve symmetry and pattern". Symmetry 2. In International Series in Modern Applied Mathematics and Computer Science. Pergamon. pp. 89–91. doi:10.1016/B978-0-08-037237-2.50013-1. ISBN 9780080372372..
Mueller, Francis J. (1965), "More on Pascal's Triangle and powers of 11", The Mathematics Teacher, 58 (5): 425–428, doi:10.5951/MT.58.5.0425, JSTOR 27957164.
Low, Leone (1966), "Even more on Pascal's Triangle and Powers of 11", The Mathematics Teacher, 59 (5): 461–463, doi:10.5951/MT.59.5.0461, JSTOR 27957385.
Fjelstad, P. (1991), "Extending Pascal's Triangle", Computers & Mathematics with Applications, 21 (9): 3, doi:10.1016/0898-1221(91)90119-O.

github.com (Global: 380^th place; English: 337^th place)

Wilmot, G.P. (2023), The Algebra Of Geometry

ias.ac.in (Global: 6,516^th place; English: 4,685^th place)

repository.ias.ac.in

Alsdorf, Ludwig (1991) [1933]. "The Pratyayas: Indian Contribution to Combinatorics" (PDF). Indian Journal of History of Science. 26 (1): 17–61. Translated by S. R. Sarma from " $π$ Die Pratyayas. Ein Beitrag zur indischen Mathematik". Zeitschrift für Indologie und Iranistik. 9: 97–157. 1933.
Bag, Amulya Kumar (1966). "Binomial theorem in ancient India" (PDF). Indian Journal of History of Science. 1 (1): 68–74.
Tertiary sources:
Sen, Samarendra Nath (1971). "Mathematics". In Bose, D. M. (ed.). A Concise History Of Science In India. Indian National Science Academy. Ch. 3, pp. 136–212, esp. "Permutations, Combinations and Pascal Triangle", pp. 156–157.
Fowler, David H. (1996). "The Binomial Coefficient Function". The American Mathematical Monthly. 103 (1): 1–17, esp. §4 "A Historical Note", pp. 10–17. doi:10.2307/2975209. JSTOR 2975209.

insa.nic.in (Global: low place; English: low place)

Alsdorf, Ludwig (1991) [1933]. "The Pratyayas: Indian Contribution to Combinatorics" (PDF). Indian Journal of History of Science. 26 (1): 17–61. Translated by S. R. Sarma from " $π$ Die Pratyayas. Ein Beitrag zur indischen Mathematik". Zeitschrift für Indologie und Iranistik. 9: 97–157. 1933.
Bag, Amulya Kumar (1966). "Binomial theorem in ancient India" (PDF). Indian Journal of History of Science. 1 (1): 68–74.
Tertiary sources:
Sen, Samarendra Nath (1971). "Mathematics". In Bose, D. M. (ed.). A Concise History Of Science In India. Indian National Science Academy. Ch. 3, pp. 136–212, esp. "Permutations, Combinations and Pascal Triangle", pp. 156–157.
Fowler, David H. (1996). "The Binomial Coefficient Function". The American Mathematical Monthly. 103 (1): 1–17, esp. §4 "A Historical Note", pp. 10–17. doi:10.2307/2975209. JSTOR 2975209.

jstor.org (Global: 23^rd place; English: 15^th place)

Coolidge, J. L. (1949), "The story of the binomial theorem", The American Mathematical Monthly, 56 (3): 147–157, doi:10.2307/2305028, JSTOR 2305028, MR 0028222.
Cobeli, Cristian; Zaharescu, Alexandru (2013). "Promenade around Pascal Triangle — Number Motives". Bulletin mathématique de la Société des Sciences Mathématiques de Roumanie. 56 (104) (1). Societatea de Științe Matematice din România: 74. JSTOR 43679285. Known for more than a millenium in Asia and Europe (cf. Burton [Bur'07]), the origins of the Pascal triangle are lost in the mist of time. In Chandahsāstra, the Hindu scholar Pingala has classified meters (chandas) or rhythm of poems that are closely allied to music (Bag [Bag'66]). He enumerated and counted the meters of a given length n that have exactly r syllables of a kind. In doing this, he obtained Meruprastāra (the stairway to the mythical mountain Meru). Then Halayudha (cca. 975) in Mṛta-Sañjīvanī, a text of commentaries on Pingala's Chandahsāstra, clearly described Meruprastāra as what is today known as the arithmetic triangle. Among those who considered the triangle before Pascal, we find: Al-Karaji (953-1029); Jia Xian (1010-1070), China; Al-Samawal al-Maghribi...
Alsdorf, Ludwig (1991) [1933]. "The Pratyayas: Indian Contribution to Combinatorics" (PDF). Indian Journal of History of Science. 26 (1): 17–61. Translated by S. R. Sarma from " $π$ Die Pratyayas. Ein Beitrag zur indischen Mathematik". Zeitschrift für Indologie und Iranistik. 9: 97–157. 1933.
Bag, Amulya Kumar (1966). "Binomial theorem in ancient India" (PDF). Indian Journal of History of Science. 1 (1): 68–74.
Tertiary sources:
Sen, Samarendra Nath (1971). "Mathematics". In Bose, D. M. (ed.). A Concise History Of Science In India. Indian National Science Academy. Ch. 3, pp. 136–212, esp. "Permutations, Combinations and Pascal Triangle", pp. 156–157.
Fowler, David H. (1996). "The Binomial Coefficient Function". The American Mathematical Monthly. 103 (1): 1–17, esp. §4 "A Historical Note", pp. 10–17. doi:10.2307/2975209. JSTOR 2975209.
Kennedy, E. (1966). Omar Khayyam. The Mathematics Teacher 1958. National Council of Teachers of Mathematics. pp. 140–142. JSTOR i27957284.
Fowler, David (January 1996). "The Binomial Coefficient Function". The American Mathematical Monthly. 103 (1): 1–17. doi:10.2307/2975209. JSTOR 2975209. See in particular p. 11.
Fine, N. J. (1947), "Binomial coefficients modulo a prime", American Mathematical Monthly, 54 (10): 589–592, doi:10.2307/2304500, JSTOR 2304500, MR 0023257. See in particular Theorem 2, which gives a generalization of this fact for all prime moduli.
Hinz, Andreas M. (1992), "Pascal's triangle and the Tower of Hanoi", The American Mathematical Monthly, 99 (6): 538–544, doi:10.2307/2324061, JSTOR 2324061, MR 1166003. Hinz attributes this observation to an 1891 book by Édouard Lucas, Théorie des nombres (p. 420).
Morton, Robert L. (1964), "Pascal's Triangle and powers of 11", The Mathematics Teacher, 57 (6): 392–394, doi:10.5951/MT.57.6.0392, JSTOR 27957091.
Winteridge, David J. (1984), "Pascal's Triangle and Powers of 11", Mathematics in School, 13 (1): 12–13, JSTOR 30213884.
Mueller, Francis J. (1965), "More on Pascal's Triangle and powers of 11", The Mathematics Teacher, 58 (5): 425–428, doi:10.5951/MT.58.5.0425, JSTOR 27957164.
Low, Leone (1966), "Even more on Pascal's Triangle and Powers of 11", The Mathematics Teacher, 59 (5): 461–463, doi:10.5951/MT.59.5.0461, JSTOR 27957385.

loc.gov (Global: 63^rd place; English: 47^th place)

Newton, Isaac (1736), "A Treatise of the Method of Fluxions and Infinite Series", The Mathematical Works of Isaac Newton: 1:31–33, But these in the alternate areas, which are given, I observed were the same with the figures of which the several ascending powers of the number 11 consist, viz. $11^{0}$ , $11^{1}$ , $11^{2}$ , $11^{3}$ , $11^{4}$ , etc. that is, first 1; the second 1, 1; the third 1, 2, 1; the fourth 1, 3, 3, 1; the fifth 1, 4, 6, 4, 1, and so on.

researchgate.net (Global: 96^th place; English: 89^th place)

Islam, Robiul; et al. (2020), Finding any row of Pascal's triangle extending the concept of power of 11.

sciencedirect.com (Global: 137^th place; English: 146^th place)

Hilton, P.; et al. (1989). "Extending the binomial coefficients to preserve symmetry and pattern". Symmetry 2. In International Series in Modern Applied Mathematics and Computer Science. Pergamon. pp. 89–102. doi:10.1016/B978-0-08-037237-2.50013-1. ISBN 9780080372372..
Hilton, P.; et al. (1989). "Extending the binomial coefficients to preserve symmetry and pattern". Symmetry 2. In International Series in Modern Applied Mathematics and Computer Science. Pergamon. pp. 100–102. doi:10.1016/B978-0-08-037237-2.50013-1. ISBN 9780080372372..
Hilton, P.; et al. (1989). "Extending the binomial coefficients to preserve symmetry and pattern". Symmetry 2. In International Series in Modern Applied Mathematics and Computer Science. Pergamon. pp. 89–91. doi:10.1016/B978-0-08-037237-2.50013-1. ISBN 9780080372372..

semanticscholar.org (Global: 15^th place; English: 8^th place)

api.semanticscholar.org

Brothers, H. J. (2012), "Finding e in Pascal's triangle", Mathematics Magazine, 85 (1): 51, doi:10.4169/math.mag.85.1.51, S2CID 218541210.
Brothers, H. J. (2012), "Pascal's triangle: The hidden stor-e", The Mathematical Gazette, 96 (535): 145–148, doi:10.1017/S0025557200004204, S2CID 233356674.

st-andrews.ac.uk (Global: 1,053^rd place; English: 1,274^th place)

mathshistory.st-andrews.ac.uk

"Varahamihira (505–587) – Biography". MacTutor History of Mathematics Archive. University of St Andrews. Retrieved 26 July 2026.

udayton.edu (Global: low place; English: 6,322^nd place)

ecommons.udayton.edu

Arnold, Robert; et al. (2004), "Newton's Unfinished Business: Uncovering the Hidden Powers of Eleven in Pascal's Triangle", Proceedings of Undergraduate Mathematics Day.

usu.edu (Global: 5,967^th place; English: 3,457^th place)

5010.mathed.usu.edu

"Pascal's Triangle in Probability". 5010.mathed.usu.edu. Retrieved 2023-06-01.

Pascal's triangle (English Wikipedia)

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sciencedirect.com (Global: 137th place; English: 146th place)

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mathshistory.st-andrews.ac.uk

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ams.org (Global: 434^th place; English: 249^th place)

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bnf.fr (Global: 61^st place; English: 357^th place)

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loc.gov (Global: 63^rd place; English: 47^th place)

researchgate.net (Global: 96^th place; English: 89^th place)

sciencedirect.com (Global: 137^th place; English: 146^th place)

semanticscholar.org (Global: 15^th place; English: 8^th place)

st-andrews.ac.uk (Global: 1,053^rd place; English: 1,274^th place)

udayton.edu (Global: low place; English: 6,322^nd place)

usu.edu (Global: 5,967^th place; English: 3,457^th place)