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The alternating factorial is the absolute value of the alternating sum of the first factorials, = ()!. These have mainly been studied in connection with their primality; only finitely many of them can be prime, but a complete list of primes of this form is not known.
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)! (its place value).
I propose to write !! for such products, and if a name be required for the product to call it the "alternate factorial" or the "double factorial". Meserve (1948) [9] states that the double factorial was originally introduced in order to simplify the expression of certain trigonometric integrals that arise in the derivation of the Wallis product.
The value of each is taken to be 1 (an empty product) when =. These symbols are collectively called factorial powers. [2] The Pochhammer symbol, introduced by Leo August Pochhammer, is the notation (), where n is a non-negative integer.
The number 7! is the lowest factorial which is multiple of 140, so the proper factor 139 is found in this step. Using another initial value, say A = 9, we get: V 1 of seq(9) = V 1! of seq(9) = 9
Table of factorial and its remainder modulo n ... 9: 40320: 0 10: 362880: 0 11: 3628800: 10 12: 39916800: 0 13: ... but they are too slow to have practical value.
(n factorial) is the number of n-permutations; !n (n subfactorial) is the number of derangements – n-permutations where all of the n elements change their initial places. In combinatorial mathematics , a derangement is a permutation of the elements of a set in which no element appears in its original position.
9: 0 +0.000000000 10 5 / ... Most striking in this context is the fact that the falling factorial c k−1 has for k = 0 the value ... The denominators of S n+1 ...