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  2. Central binomial coefficient - Wikipedia

    en.wikipedia.org/wiki/Central_binomial_coefficient

    The central binomial coefficients give the number of possible number of assignments of n -a-side sports teams from 2 n players, taking into account the playing area side. The central binomial coefficient is the number of arrangements where there are an equal number of two types of objects. For example, when , the binomial coefficient is equal ...

  3. Binomial theorem - Wikipedia

    en.wikipedia.org/wiki/Binomial_theorem

    In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, it is possible to expand the polynomial (x + y) n into a sum involving terms of the form ax b y c, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending ...

  4. Catalan number - Wikipedia

    en.wikipedia.org/wiki/Catalan_number

    Given a polygon P with n + 2 sides and a triangulation, mark one of its sides as the base, and also orient one of its 2n + 1 total edges. There are (4n + 2)C n such marked triangulations for a given base. Given a polygon Q with n + 3 sides and a (different) triangulation, again mark one of its sides as the base. Mark one of the sides other than ...

  5. 1 + 2 + 3 + 4 + ⋯ - Wikipedia

    en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E...

    The partial sums of the series 1 + 2 + 3 + 4 + 5 + 6 + ⋯ are 1, 3, 6, 10, 15, etc.The nth partial sum is given by a simple formula: = = (+). This equation was known ...

  6. 1/2 + 1/4 + 1/8 + 1/16 + ⋯ - Wikipedia

    en.wikipedia.org/wiki/1/2_%2B_1/4_%2B_1/8_%2B_1/...

    1/2 + 1/4 + 1/8 + 1/16 + ⋯. First six summands drawn as portions of a square. The geometric series on the real line. In mathematics, the infinite series ⁠ 1 2 ⁠ + ⁠ 1 4 ⁠ + ⁠ 1 8 ⁠ + ⁠ 1 16 ⁠ + ··· is an elementary example of a geometric series that converges absolutely. The sum of the series is 1. In summation notation ...

  7. Particular values of the Riemann zeta function - Wikipedia

    en.wikipedia.org/wiki/Particular_values_of_the...

    It is known that ζ(3) is irrational (Apéry's theorem) and that infinitely many of the numbers ζ(2n + 1) : n ∈ , are irrational. There are also results on the irrationality of values of the Riemann zeta function at the elements of certain subsets of the positive odd integers; for example, at least one of ζ (5), ζ (7), ζ (9), or ζ (11 ...

  8. 1 − 2 + 3 − 4 + ⋯ - Wikipedia

    en.wikipedia.org/wiki/1_%E2%88%92_2_%2B_3_%E2%88...

    The idea becomes clearer by considering the general series 1 − 2x + 3x 2 − 4x 3 + 5x 4 − 6x 5 + &c. that arises while expanding the expression 1 ⁄ (1+x) 2, which this series is indeed equal to after we set x = 1.

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