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For example, 9!! = 1 × 3 × 5 × 7 × 9 = 945. Double factorials are used in trigonometric integrals, [92] in expressions for the gamma function at half-integers and the volumes of hyperspheres, [93] and in counting binary trees and perfect matchings. [91] [94] Exponential factorial
The factorial of a positive n may be written as the product of two double factorials: [3]! =!! ...
The term "factorial number system" is used by Knuth, [3] while the French equivalent "numération factorielle" was first used in 1888. [4] The term "factoradic", which is a portmanteau of factorial and mixed radix, appears to be of more recent date.
An alternative notation for the rising factorial () is the less common () +. When () + is used to denote the rising factorial, the notation () is typically used for the ordinary falling factorial, to avoid confusion. [3]
In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of . It is named after James Stirling, though a related but less precise result was first stated by Abraham de Moivre. [1] [2] [3]
2.3 Trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions relationship 2.4 Modified-factorial denominators 2.5 Binomial coefficients
3. Essential items for a relaxing activity/workout. 4. The words in this category precede a three-letter noun (hint: the noun typically refers to a lid or cover for something).
The factorial is defined by the recurrence relation ! = ... there is also a nice method to solve it: [3] + ...