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  2. Exponential factorial - Wikipedia

    en.wikipedia.org/wiki/Exponential_factorial

    The exponential factorial is a positive integer n raised to the power of n − 1, ... The first few exponential factorials are 1, 2, 9, 262144, ...

  3. Factorial - Wikipedia

    en.wikipedia.org/wiki/Factorial

    The exponential factorial is defined recursively as =, =. For example, the exponential factorial of 4 is 4 3 2 1 = 262144. {\displaystyle 4^{3^{2^{1}}}=262144.} These numbers grow much more quickly than regular factorials.

  4. List of mathematical series - Wikipedia

    en.wikipedia.org/wiki/List_of_mathematical_series

    2.2 Exponential function 2.3 Trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions relationship 2.4 Modified-factorial denominators

  5. Exponential function - Wikipedia

    en.wikipedia.org/wiki/Exponential_function

    The exponential function is the sum of a power series: [2] [3] ⁡ = + +! +! + = =!, where ! is the factorial of n (the product of the n first positive integers). This series is absolutely convergent for every x {\displaystyle x} per the ratio test .

  6. List of sums of reciprocals - Wikipedia

    en.wikipedia.org/wiki/List_of_sums_of_reciprocals

    An exponential factorial is an operation recursively defined as =, = . For example, a 4 = 4 3 2 1 {\displaystyle \ a_{4}=4^{3^{2^{1}}}\ } where the exponents are evaluated from the top down. The sum of the reciprocals of the exponential factorials from 1 onward is approximately 1.6111 and is transcendental.

  7. Stirling numbers of the first kind - Wikipedia

    en.wikipedia.org/wiki/Stirling_numbers_of_the...

    In particular, the unsigned Stirling numbers of the first kind count permutations according to their number of cycles (counting fixed points as cycles of length one). [1] The Stirling numbers of the first and second kind can be understood as inverses of one another when viewed as triangular matrices. This article is devoted to specifics of ...

  8. History of mathematical notation - Wikipedia

    en.wikipedia.org/wiki/History_of_mathematical...

    The first formulation of a quantum theory describing radiation and matter interaction is due to Paul Adrien Maurice Dirac, who, during 1920, was first able to compute the coefficient of spontaneous emission of an atom. [99] In 1928, the relativistic Dirac equation was formulated by Dirac to explain the behavior of the relativistically moving ...

  9. Euler's totient function - Wikipedia

    en.wikipedia.org/wiki/Euler's_totient_function

    The first thousand values of φ(n).The points on the top line represent φ(p) when p is a prime number, which is p − 1. [1]In number theory, Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n.