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In mathematics, Pascal's triangle is a triangular array of the binomial coefficients. In much of the Western world, it is named after French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India,〔Maurice Winternitz, ''History of Indian Literature'', Vol. III〕 Iran, China, Germany, and Italy. The rows of Pascal's triangle are conventionally enumerated starting with row ''n'' = 0 at the top (the 0th row). The entries in each row are numbered from the left beginning with ''k'' = 0 and are usually staggered relative to the numbers in the adjacent rows. Having the indices of both rows and columns start at zero makes it possible to state that the binomial coefficient appears in the ''n''th row and ''k''th column of Pascal's triangle. A simple construction of the triangle proceeds in the following manner: In row 0, the topmost row, the entry is (the entry is in the zeroth row and zeroth column). Then, to construct the elements of the following rows, add the number above and to the left with the number above and to the right of a given position to find the new value to place in that position. If either the number to the right or left is not present, substitute a zero in its place. For example, the initial number in the first (or any other) row is 1 (the sum of 0 and 1), whereas the numbers 1 and 3 in the third row are added to produce the number 4 in the fourth row. This construction is related to the binomial coefficients by Pascal's rule, which says that if : then : for any non-negative integer ''n'' and any integer ''k'' between 0 and ''n''.〔The binomial coefficient is conventionally set to zero if ''k'' is either less than zero or greater than ''n'' .〕 Pascal's triangle has higher dimensional generalizations. The three-dimensional version is called ''Pascal's pyramid'' or ''Pascal's tetrahedron'', while the general versions are called ''Pascal's simplices''. ==History== The pattern of numbers that forms Pascal's triangle was known well before Pascal's time. Pascal innovated many previously unattested uses of the triangle's numbers, uses he described comprehensively in what is perhaps the earliest known mathematical treatise to be specially devoted to the triangle, his ''Traité du triangle arithmétique'' (1653). Centuries before, discussion of the numbers had arisen in the context of Indian studies of combinatorics and of binomial numbers and Greeks' study of figurate numbers.〔(Pascal's Triangle | World of Mathematics Summary )〕 From later commentary, it appears that the binomial coefficients and the additive formula for generating them, , were known to Pingala in or before the 2nd century BC.〔A. W. F. Edwards. ''Pascal's arithmetical triangle: the story of a mathematical idea.'' JHU Press, 2002. Pages 30–31.〕〔.〕 While Pingala's work only survives in fragments, the commentator Varāhamihira, around 505, gave a clear description of the additive formula,〔 and a more detailed explanation of the same rule was given by Halayudha, around 975. Halayudha also explained obscure references to ''Meru-prastaara'', the "Staircase of Mount Meru", giving the first surviving description of the arrangement of these numbers into a triangle.〔 In approximately 850, the Jain mathematician Mahāvīra gave a different formula for the binomial coefficients, using multiplication, equivalent to the modern formula .〔 In 1068, four columns of the first sixteen rows were given by the mathematician Bhattotpala, who was the first recorded mathematician to equate the additive and multiplicative formulas for these numbers.〔 At around the same time, it was discussed in Persia (Iran) by the Persian mathematician, Al-Karaji (953–1029). It was later repeated by the Persian poet-astronomer-mathematician Omar Khayyám (1048–1131); thus the triangle is referred to as the Khayyam-Pascal triangle or Khayyam triangle in Iran. Several theorems related to the triangle were known, including the binomial theorem. Khayyam used a method of finding ''n''th roots based on the binomial expansion, and therefore on the binomial coefficients.〔.〕 Pascal's triangle was known in China in the early 11th century through the work of the Chinese mathematician Jia Xian (1010–1070). In the 13th century, Yang Hui (1238–1298) presented the triangle and hence it is still called Yang Hui's triangle in China.〔Weisstein, Eric W. (2003). ''CRC concise encyclopedia of mathematics'', p.2169. ISBN 978-1-58488-347-0.〕 In the west, the binomial coefficients were calculated by Gersonides in the early 14th century, using the multiplicative formula for them.〔 Petrus Apianus (1495–1552) published the full triangle on the frontispiece of his book on business calculations in 1527. This is the first record of the triangle in Europe.〔.〕 Michael Stifel published a portion of the triangle (from the second to the middle column in each row) in 1544, describing it as a table of figurate numbers.〔 In Italy, Pascal's triangle is referred to as Tartaglia's triangle, named for the Italian algebraist Niccolò Fontana Tartaglia (1500–1577), who published six rows of the triangle in 1556.〔 Gerolamo Cardano, also, published the triangle as well as the additive and multiplicative rules for constructing it in 1570.〔 Pascal's ''Traité du triangle arithmétique'' (''Treatise on Arithmetical Triangle'') was published posthumously in 1665. In this, Pascal collected several results then known about the triangle, and employed them to solve problems in probability theory. The triangle was later named after Pascal by Pierre Raymond de Montmort (1708) who called it "Table de M. Pascal pour les combinaisons" (French: Table of Mr. Pascal for combinations) and Abraham de Moivre (1730) who called it "Triangulum Arithmeticum PASCALIANUM" (Latin: Pascal's Arithmetic Triangle), which became the modern Western name.〔 See in particular p. 11.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Pascal's triangle」の詳細全文を読む スポンサード リンク
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