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The factorial of is , or in symbols, ! =. There are several motivations for this definition: For n = 0 {\displaystyle n=0} , the definition of n ! {\displaystyle n!} as a product involves the product of no numbers at all, and so is an example of the broader convention that the empty product , a product of no factors, is equal to the ...
Stirling permutations, permutations of the multiset of numbers 1, 1, 2, 2, ..., k, k in which each pair of equal numbers is separated only by larger numbers, where k = n + 1 / 2 . The two copies of k must be adjacent; removing them from the permutation leaves a permutation in which the maximum element is k − 1 , with n positions into ...
In logic, a set of symbols is commonly used to express logical representation. The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics.
1. Factorial: if n is a positive integer, n! is the product of the first n positive integers, and is read as "n factorial". 2. Double factorial: if n is a positive integer, n!! is the product of all positive integers up to n with the same parity as n, and is read as "the double factorial of n". 3.
These symbols are collectively called factorial powers. [ 2 ] The Pochhammer symbol , introduced by Leo August Pochhammer , is the notation ( x ) n {\displaystyle (x)_{n}} , where n is a non-negative integer .
As one special case, it can be used to prove that if n is a positive integer then 4 divides () if and only if n is not a power of 2. It follows from Legendre's formula that the p -adic exponential function has radius of convergence p − 1 / ( p − 1 ) {\displaystyle p^{-1/(p-1)}} .
The factorial number system is a mixed radix numeral system: the i-th digit from the right has base i, which means that the digit must be strictly less than i, and that (taking into account the bases of the less significant digits) its value is to be multiplied by (i − 1)!
Also note the keyboard had some 55 unique (68 listed per tables above, including comparative symbols but several symbols appear in both monadic and dyadic tables) APL symbol keys (55 APL functions (operators) are listed in IBM's 5110 APL Reference Manual), thus with the use of alt, shift and ctrl keys - it would theoretically have allowed a ...