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These are counted by the double factorial 15 = (6 − 1)‼. In mathematics, the double factorial of a number n, denoted by n‼, is the product of all the positive integers up to n that have the same parity (odd or even) as n. [1] That is,
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
It is unknown whether these constants are transcendental in general, but Γ( 1 / 3 ) and Γ( 1 / 4 ) were shown to be transcendental by G. V. Chudnovsky. Γ( 1 / 4 ) / 4 √ π has also long been known to be transcendental, and Yuri Nesterenko proved in 1996 that Γ( 1 / 4 ), π, and e π are algebraically independent.
In the previous two integrals, n!! is the double factorial: for even n it is equal to the product of all even numbers from 2 to n, and for odd n it is the product of all odd numbers from 1 to n; additionally it is assumed that 0!! = (−1)!! = 1.
The number of perfect matchings of the complete graph K n (with n even) is given by the double factorial (n – 1)!!. [ 12 ] The crossing numbers up to K 27 are known, with K 28 requiring either 7233 or 7234 crossings.
(), where !! is the notation for the double factorial. [4] The hyperfactorials give the sequence of discriminants of Hermite polynomials in ... [1] References ...
[1] [2] [3] One way of stating the approximation involves the logarithm of the factorial: (!) = + (), where the big O notation means that, for all sufficiently large values of , the difference between (!
34,459,425 = double factorial of 17 34,012,224 = 5832 2 = 324 3 = 18 6 34,636,834 = number of 31-bead binary necklaces with beads of 2 colors where the colors may be swapped but turning over is not allowed [ 15 ]