The double factorial, denoted by the symbol "!!" or "n!!", is a mathematical operation that is used to find the product of all the positive integers that are less than or equal to a given integer, but with certain elements removed from the product. Specifically, the double factorial only includes every other integer in the product, starting with the given integer and working backwards. For example, the double factorial of 6, written as 6!!, is equal to 6 * 4 * 2, or 48.

The double factorial can be defined for any positive integer n as follows:

n!! = n * (n-2) * (n-4) * ... * 2 if n is even
n!! = n * (n-2) * (n-4) * ... * 1 if n is odd

Note that if n is even, the double factorial includes every other integer from n down to 2, while if n is odd, it includes every other integer from n down to 1.

One way to understand the double factorial is to think of it as a shortened version of the factorial operation, which is denoted by the symbol "!". The factorial of a positive integer n, written as n!, is equal to the product of all the positive integers from 1 to n. For example, the factorial of 6, written as 6!, is equal to 6 * 5 * 4 * 3 * 2 * 1, or 720.

The double factorial can be thought of as a shortened version of the factorial operation because it only includes certain integers in the product. For example, the double factorial of 6, 6!!, is equal to 6 * 4 * 2, or 48, which is a shortened version of the factorial of 6, 6!, which is equal to 6 * 5 * 4 * 3 * 2 * 1, or 720.

The double factorial is useful in a number of mathematical contexts, including combinatorics, probability theory, and mathematical analysis. It arises frequently in the study of combinatorial objects such as permutations and combinations, and it is also used in the calculation of certain probability distributions, such as the negative binomial distribution and the Poisson distribution. In mathematical analysis, the double factorial is used in the study of special functions, such as the gamma function and the Bessel functions.

In conclusion, the double factorial is a mathematical operation that is used to find the product of certain integers, with certain elements removed from the product. It is a shortened version of the factorial operation and is useful in a number of mathematical contexts, including combinatorics, probability theory, and mathematical analysis.

## Double factorial

For instance, a complete graph with four vertices a, b, c, and d has three perfect matchings: ab and cd, ac and bd, and ad and bc. The solution is to define the double factorial using gamma function. } Similar identities can be obtained via context-free grammars. This case is similar to the unrooted case, but the number of edges that can be subdivided is even, and in addition to subdividing an edge it is possible to add a node to a tree with one fewer leaf by adding a new root whose two children are the smaller tree and the new leaf. EATCS Monographs in Theoretical Computer Science. For example, the Stirling permutations of {1,1,2,2} are those where no 1 appears between the 2s. Provide details and share your research! See in particular p.

## Double Factorial

The factorial of a non-zero n may be written as the product of two double factorials: n! For odd numbers, n!! In such a graph, any single vertex v has n possible choices of vertex that it can be matched to, and once this choice is made the remaining problem is one of selecting a perfect matching in a complete graph with two fewer vertices. Proceedings of the Royal Society of London. However I wrote this code in Python 3. I propose to write n! The rest of them, {1,2,1,2}, {2,1,1,2}, and {2,1,2,1}, all have at least one 1 between the pair of 2s. Journal of Statistical Planning and Inference.

## Multifactorial Calculator

Similarly, {1,1,2,2,3,3}, {1,1,2,3,3,2}, {1,3,3,2,2,1}, are examples that meet the condition, but {3,2,2,3,1,1}, {1,2,3,1,2,3}, and {1,2,3,2,3,1} and others are not. Thanks for contributing an answer to Stack Overflow! } for such products, and if a name be required for the product to call it the "alternate factorial" or the "double factorial. One way to visualize Stirling permutations is as a simple list plot. Below is the implementation: Give the number as user input using the int input function and store it in a variable. In the previous article, we have discussed Factorial: The product of all positive integers less than or equal to n is the factorial of a non-negative integer n, denoted by n! } A different extension of the double factorial, which agrees with the formula z! Below is the implementation: Give the number as static input and store it in a variable. I've managed to do a single recursive factorial. I've been stucked on this question for a really long time.