The Mathcounts competitors set a Guinness World Record for the fastest human formation of the Pascal Triangle.
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An example of a weakly unimodal function which is not strongly unimodal is every other row in a Pascal triangle.
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For instance, the Pascal triangle, with the nodes connected by appropriate downward arrows, is the Bratteli diagram of an AF algebra.
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To do this, he makes use of the Pascal triangle, which he labels as the diagram of an ancient method first discovered by Jia Xian before 1050.
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What I am looking for is some kind of generating function in terms of the coordinates of the number, ( similar to the relationship of binomial coefficients to pascals triangle ).
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The rows of the ( 2, 1 )-Pascal triangle are conventionally enumerated starting with row " n " = 0 at the top ( the 0th row ).
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Zhu also solved square and cube roots problems by solving quadratic and cubic equations, and added to the understanding of series and progressions, classifying them according to the coefficients of the Pascal triangle.
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Given the parameters of a Steiner system and a subset of size t'\ leq t, contained in at least one block, one can compute the number of blocks intersecting that subset in a fixed number of elements by constructing a Pascal triangle.
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Sometimes there's an interesting pattern, like you can generate a sierpinski triangle with a modulo on a pascal triangle, or a harter-heighway dragon with a series of " folding " operations on a list of turns, but these, and a great deal of fractals fall into the category that can be produced by recursive copies.
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Another relationship involves the Pascal Triangle : Whereas the classical Pascal Triangle with sides has diagonals with the natural numbers, triangular numbers, and tetrahedral numbers, generating the Fibonacci numbers as sums of samplings across diagonals, the sister Pascal with sides has equivalent diagonals with odd numbers, square numbers, and square pyramidal numbers, respectively, and generates ( by the same procedure ) the Lucas numbers rather than Fibonacci.
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