Factorial

Selected members of the factorial sequence Template:OEIS; values specified in scientific notation are rounded to the displayed precision

Template:Math	Template:Math
0	1
1	1
2	2
3	6
4	24
5	120
6	720
7	Template:Val
8	Template:Val
9	Template:Val
10	Template:Val
11	Template:Val
12	Template:Val
13	Template:Val
14	Template:Val
15	Template:Val
16	Template:Val
17	Template:Val
18	Template:Val
19	Template:Val
20	Template:Val
25	Template:Val
50	Template:Val
70	Template:Val
100	Template:Val
450	Template:Val
Template:Val	Template:Val
Template:Val	Template:Val
Template:Val	Template:Val
Template:Val	Template:Val
Template:Val	Template:Val
Template:Val	Template:Val
Template:Val	Template:Val
[[googol|Template:Val]]	10Template:Val

In mathematics, the factorial of a positive integer Template:Mvar, denoted by Template:Math, is the product of all positive integers less than or equal to Template:Mvar:

$n!=n\times (n-1)\times (n-2)\times (n-3)\times \cdots \times 3\times 2\times 1\,.$

For example,

$5!=5\times 4\times 3\times 2\times 1=120\,.$

The value of 0! is 1, according to the convention for an empty product.Template:Sfn

The factorial operation is encountered in many areas of mathematics, notably in combinatorics, algebra, and mathematical analysis. Its most basic use counts the possible distinct sequences – the permutations – of Template:Mvar distinct objects: there are Template:Math.

The factorial function can also be extended to non-integer arguments while retaining its most important properties. This involves using gamma function to define Template:Math. However, this extension does not work when Template:Mvar is a negative integer.

History

Factorials were used to count permutations at least as early as the 12th century, by Indian scholars. In 1677, Fabian Stedman described factorials as applied to change ringing, a musical art involving the ringing of many tuned bells.Template:Sfn After describing a recursive approach, Stedman gives a statement of a factorial (using the language of the original): Template:Cquote The notation Template:Math was introduced by the French mathematician Christian Kramp in 1808.

Definition

The factorial function is defined by the product

$n!=1\cdot 2\cdot 3\cdots (n-2)\cdot (n-1)\cdot n,$

for integer Template:Math. This may be written in pi product notation as

$n!=\prod _{i=1}^{n}i.$

From these formulas, one may derive the recurrence relation

$n!=n\cdot (n-1)!.$

For example, one has

${\begin{aligned}5!&=5\cdot 4!\\6!&=6\cdot 5!\\50!&=50\cdot 49!\end{aligned}}$

and so on.

Factorial of zero

The factorial of Template:Math is Template:Math, or in symbols, Template:Math.

There are several motivations for this definition:

• For Template:Math, the definition of Template:Math as a product involves the product of no numbers at all, and so is an example of the broader convention that the product of no factors is equal to the multiplicative identity (see Empty product).
• There is exactly one permutation of zero objects (with nothing to permute, the only rearrangement is to do nothing).
• It makes many identities in combinatorics valid for all applicable sizes. The number of ways to choose 0 elements from the empty set is given by the binomial coefficient

${\binom {0}{0}}={\frac {0!}{0!0!}}=1.$

More generally, the number of ways to choose all Template:Mvar elements among a set of Template:Mvar is

${\binom {n}{n}}={\frac {n!}{n!0!}}=1.$

• It allows for the compact expression of many formulae, such as the exponential function, as a power series:

$e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}.$

• It extends the recurrence relation to 0.

Applications

Although the factorial function has its roots in combinatorics, formulas involving factorials occur in many areas of mathematics.

• There are Template:Math different ways of arranging Template:Mvar distinct objects into a sequence, the permutations of those objects.
• Often factorials appear in the denominator of a formula to account for the fact that ordering is to be ignored. A classical example is counting Template:Mvar-combinations (subsets of Template:Mvar elements) from a set with Template:Mvar elements. One can obtain such a combination by choosing a Template:Mvar-permutation: successively selecting and removing one element of the set, Template:Mvar times, for a total of

$(n-0)(n-1)(n-2)\cdots \left(n-(k-1)\right)={\frac {n!}{(n-k)!}}=n^{\underline {k}}$

possibilities. This, however, produces the Template:Mvar-combinations in a particular order that one wishes to ignore; since each Template:Mvar-combination is obtained in Template:Math different ways, the correct number of Template:Mvar-combinations is

${\frac {n(n-1)(n-2)\cdots (n-k+1)}{k(k-1)(k-2)\cdots 1}}={\frac {n^{\underline {k}}}{k!}}={\frac {n!}{(n-k)!k!}}={\binom {n}{k}}.$

This number is known as the binomial coefficient, because it is also the coefficient of Template:Math in Template:Math. The term $n^{\underline {k}}$ is often called a falling factorial (pronounced "n to the falling k").

• Factorials occur in algebra for various reasons, such as via the already mentioned coefficients of the binomial formula, or through averaging over permutations for symmetrization of certain operations.
• Factorials also turn up in calculus; for example, they occur in the denominators of the terms of Taylor's formula, where they are used as compensation terms due to the Template:Mvarth derivative of Template:Math being equivalent to Template:Math.
• Factorials are also used extensively in probability theory and number theory (see below).
• Factorials can be useful to facilitate expression manipulation. For instance the number of Template:Mvar-permutations of Template:Mvar can be written as

$n^{\underline {k}}={\frac {n!}{(n-k)!}}\,;$

while this is inefficient as a means to compute that number, it may serve to prove a symmetry property of binomial coefficients:

${\binom {n}{k}}={\frac {n^{\underline {k}}}{k!}}={\frac {n!}{(n-k)!k!}}={\frac {n^{\underline {n-k}}}{(n-k)!}}={\binom {n}{n-k}}\,.$

• The factorial function can be shown, using the power rule, to be

$n!=D^{n}\,x^{n}={\frac {d^{n}}{dx^{n}}}\,x^{n}$

where Template:Math is Euler's notation for the Template:Mvarth derivative of Template:Math.

Rate of growth and approximations for large Template:Mvar

Plot of the natural logarithm of the factorial

As Template:Mvar grows, the factorial Template:Math increases faster than all polynomials and exponential functions (but slower than $n^{n}$and double exponential functions) in Template:Mvar.

Most approximations for n! are based on approximating its natural logarithm

$\ln n!=\sum _{x=1}^{n}\ln x\,.$

The graph of the function Template:Math is shown in the figure on the right. It looks approximately linear for all reasonable values of Template:Mvar, but this intuition is false. We get one of the simplest approximations for Template:Math by bounding the sum with an integral from above and below as follows:

$\int _{1}^{n}\ln x\,dx\leq \sum _{x=1}^{n}\ln x\leq \int _{0}^{n}\ln(x+1)\,dx$

which gives us the estimate

$n\ln \left({\frac {n}{e}}\right)+1\leq \ln n!\leq (n+1)\ln \left({\frac {n+1}{e}}\right)+1\,.$

Hence Template:Math (see [[Big O notation#Family of Bachmann–Landau notations|Big Template:Mvar notation]]). This result plays a key role in the analysis of the computational complexity of sorting algorithms (see comparison sort). From the bounds on Template:Math deduced above we get that

$\left({\frac {n}{e}}\right)^{n}e\leq n!\leq \left({\frac {n+1}{e}}\right)^{n+1}e\,.$

It is sometimes practical to use weaker but simpler estimates. Using the above formula it is easily shown that for all Template:Mvar we have Template:Math, and for all Template:Math we have Template:Math.

Comparison of Stirling's approximation with the factorial

For large Template:Mvar we get a better estimate for the number Template:Math using Stirling's approximation:

$n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\,.$

This in fact comes from an asymptotic series for the logarithm, and Template:Mvar factorial lies between this and the next approximation:

${\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}<n!<{\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}e^{1/(12n)}\,.$

Another approximation for Template:Math is given by Srinivasa Ramanujan Template:Harv

${\begin{aligned}\ln n!&\approx n\ln n-n+{\frac {\ln {\Bigl (}n{\bigl (}1+4n(1+2n){\bigr )}{\Bigr )}}{6}}+{\frac {\ln \pi }{2}}\\[6px]\Longrightarrow \;n!&\approx {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\left(1+{\frac {1}{2n}}+{\frac {1}{8n^{2}}}\right)^{1/6}\,.\end{aligned}}$

Both this and Stirling's approximation give a relative error on the order of Template:Math, but Ramanujan's is about four times more accurate. However, if we use two correction terms in a Stirling-type approximation, as with Ramanujan's approximation, the relative error will be of order Template:Math:Template:Citation needed

$n!\approx {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\exp \left({{\frac {1}{12n}}-{\frac {1}{360n^{3}}}}\right)\,.$

Computation

If efficiency is not a concern, computing factorials is trivial from an algorithmic point of view: successively multiplying a variable initialized to 1 by the integers up to Template:Mvar (if any) will compute Template:Math, provided the result fits in the variable. In functional languages, the recursive definition is often implemented directly to illustrate recursive functions.

The main practical difficulty in computing factorials is the size of the result. To assure that the exact result will fit for all legal values of even the smallest commonly used integral type (8-bit signed integers) would require more than 700 bits, so no reasonable specification of a factorial function using fixed-size types can avoid questions of overflow. The values 12! and 20! are the largest factorials that can be stored in, respectively, the 32-bit and 64-bit integers commonly used in personal computers, however many languages support variable length integer types capable of calculating very large values. Floating-point representation of an approximated result allows going a bit further, but this also remains quite limited by possible overflow. Most calculators use scientific notation with 2-digit decimal exponents, and the largest factorial that fits is then 69!, because Template:Nowrap. Other implementations (such as computer software such as spreadsheet programs) can often handle larger values.

Most software applications will compute small factorials by direct multiplication or table lookup. Larger factorial values can be approximated using Stirling's formula. Wolfram Alpha can calculate exact results for the ceiling function and floor function applied to the binary, natural and common logarithm of Template:Math for values of Template:Mvar up to Template:Val, and up to Template:Val for the integers.

If the exact values of large factorials are needed, they can be computed using arbitrary-precision arithmetic. Instead of doing the sequential multiplications Template:Nowrap, a program can partition the sequence into two parts, whose products are roughly the same size, and multiply them using a divide-and-conquer method. This is often more efficient.

The asymptotically best efficiency is obtained by computing Template:Math from its prime factorization. As documented by Peter Borwein, prime factorization allows Template:Math to be computed in time Template:Math, provided that a fast multiplication algorithm is used (for example, the Schönhage–Strassen algorithm). Peter Luschny presents source code and benchmarks for several efficient factorial algorithms, with or without the use of a prime sieve.

Number theory

Factorials have many applications in number theory. In particular, Template:Math is necessarily divisible by all prime numbers up to and including Template:Mvar. As a consequence, Template:Math is a composite number if and only if

$(n-1)!\equiv 0{\pmod {n}}.$

A stronger result is Wilson's theorem, which states that

$(p-1)!\equiv -1{\pmod {p}}$

if and only if Template:Mvar is prime.

Legendre's formula gives the multiplicity of the prime Template:Mvar occurring in the prime factorization of Template:Math as

$\sum _{i=1}^{\infty }\left\lfloor {\frac {n}{p^{i}}}\right\rfloor$

or, equivalently,

${\frac {n-s_{p}(n)}{p-1}},$

where Template:Math denotes the sum of the standard base-Template:Mvar digits of Template:Mvar.

Adding 1 to a factorial Template:Math yields a number that is only divisible by primes that are larger than Template:Mvar. This fact can be used to prove Euclid's theorem that the number of primes is infinite.Template:Sfn Primes of the form Template:Math are called factorial primes.

Series of reciprocals

The reciprocals of factorials produce a convergent series whose sum is [[e (mathematical constant)|the exponential base Template:Mvar]]:

$\sum _{n=0}^{\infty }{\frac {1}{n!}}={\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{6}}+{\frac {1}{24}}+{\frac {1}{120}}+\cdots =e\,.$

Although the sum of this series is an irrational number, it is possible to multiply the factorials by positive integers to produce a convergent series with a rational sum:

$\sum _{n=0}^{\infty }{\frac {1}{(n+2)n!}}={\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{8}}+{\frac {1}{30}}+{\frac {1}{144}}+\cdots =1\,.$

The convergence of this series to 1 can be seen from the fact that its partial sums are less than one by an inverse factorial. Therefore, the factorials do not form an irrationality sequence.Template:Sfn

Factorial of non-integer values

The gamma and pi functions

The gamma function interpolates the factorial function to non-integer values. The main clue is the recurrence relation generalized to a continuous domain.

Besides nonnegative integers, the factorial can also be defined for non-integer values, but this requires more advanced tools from mathematical analysis.

One function that fills in the values of the factorial (but with a shift of 1 in the argument), that is often used, is called the gamma function, denoted Template:Math. It is defined for all complex numbers Template:Mvar except for the non-positive integers, and given when the real part of Template:Mvar is positive by

$\Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}\,dt.$

Its relation to the factorial is that Template:Math for every nonnegative integer Template:Mvar.

Euler's original formula for the gamma function was

$\Gamma (z)=\lim _{n\to \infty }{\frac {n^{z}n!}{\displaystyle \prod _{k=0}^{n}(z+k)}}.$

Carl Friedrich Gauss used the notation Template:Math to denote the same function, but with argument shifted by 1, so that it agrees with the factorial for nonnegative integers. This pi function is defined by

$\Pi (z)=\int _{0}^{\infty }t^{z}e^{-t}\,dt.$

The pi function and gamma function are related by the formula Template:Math. Likewise, Template:Math for any nonnegative integer Template:Mvar.

The factorial function, generalized to all real numbers except negative integers. For example, Template:Nowrap, Template:Nowrap, Template:Nowrap.

In addition to this, the pi function satisfies the same recurrence as factorials do, but at every complex value Template:Mvar where it is defined

$\Pi (z)=z\Pi (z-1)\,.$

In fact, this is no longer a recurrence relation but a functional equation. Expressed in terms of the gamma function this functional equation takes the form

$\Gamma (n+1)=n\Gamma (n)\,.$

The values of these functions at half-integer values is therefore determined by a single one of them; one has

$\Gamma \left({\frac {1}{2}}\right)=\left(-{\frac {1}{2}}\right)!=\Pi \left(-{\frac {1}{2}}\right)={\sqrt {\pi }}\,,$

from which it follows that for Template:Math,

${\begin{aligned}&\Gamma \left({\frac {1}{2}}+n\right)=\left(-{\frac {1}{2}}+n\right)!=\Pi \left(-{\frac {1}{2}}+n\right)\\[5pt]={}&{\sqrt {\pi }}\prod _{k=1}^{n}{\frac {2k-1}{2}}={\frac {(2n)!}{4^{n}n!}}{\sqrt {\pi }}={\frac {(2n-1)!}{2^{2n-1}(n-1)!}}{\sqrt {\pi }}\,.\end{aligned}}$

For example,

$\Gamma \left({\frac {9}{2}}\right)={\frac {7}{2}}!=\Pi \left({\frac {7}{2}}\right)={\frac {1}{2}}\cdot {\frac {3}{2}}\cdot {\frac {5}{2}}\cdot {\frac {7}{2}}{\sqrt {\pi }}={\frac {8!}{4^{4}4!}}{\sqrt {\pi }}={\frac {7!}{2^{7}3!}}{\sqrt {\pi }}={\frac {105}{16}}{\sqrt {\pi }}\approx 11.631\,728\ldots$

It also follows that for Template:Math,

$\Gamma \left({\frac {1}{2}}-n\right)=\left(-{\frac {1}{2}}-n\right)!=\Pi \left(-{\frac {1}{2}}-n\right)={\sqrt {\pi }}\prod _{k=1}^{n}{\frac {2}{1-2k}}={\frac {\left(-4\right)^{n}n!}{(2n)!}}{\sqrt {\pi }}\,.$

For example,

$\Gamma \left(-{\frac {5}{2}}\right)=\left(-{\frac {7}{2}}\right)!=\Pi \left(-{\frac {7}{2}}\right)={\frac {2}{-1}}\cdot {\frac {2}{-3}}\cdot {\frac {2}{-5}}{\sqrt {\pi }}={\frac {\left(-4\right)^{3}3!}{6!}}{\sqrt {\pi }}=-{\frac {8}{15}}{\sqrt {\pi }}\approx -0.945\,308\ldots$

The pi function is certainly not the only way to extend factorials to a function defined at almost all complex values, and not even the only one that is analytic wherever it is defined. Nonetheless it is usually considered the most natural way to extend the values of the factorials to a complex function. For instance, the Bohr–Mollerup theorem states that the gamma function is the only function that takes the value 1 at 1, satisfies the functional equation Template:Math, is meromorphic on the complex numbers, and is log-convex on the positive real axis. A similar statement holds for the pi function as well, using the Template:Math functional equation.

However, there exist complex functions that are probably simpler in the sense of analytic function theory and which interpolate the factorial values. For example, Hadamard's 'gamma' function Template:Harv which, unlike the gamma function, is an entire function.

Euler also developed a convergent product approximation for the non-integer factorials, which can be seen to be equivalent to the formula for the gamma function above:

${\begin{aligned}n!=\Pi (n)&=\prod _{k=1}^{\infty }\left({\frac {k+1}{k}}\right)^{n}\!\!{\frac {k}{n+k}}\\&=\left[\left({\frac {2}{1}}\right)^{n}{\frac {1}{n+1}}\right]\left[\left({\frac {3}{2}}\right)^{n}{\frac {2}{n+2}}\right]\left[\left({\frac {4}{3}}\right)^{n}{\frac {3}{n+3}}\right]\cdots \end{aligned}}$

However, this formula does not provide a practical means of computing the pi function or the gamma function, as its rate of convergence is slow.

Applications of the gamma function

The volume of an [[Dimension|Template:Mvar-dimensional]] hypersphere of radius Template:Mvar is

$V_{n}={\frac {\pi ^{n/2}}{\Gamma \left({\frac {n}{2}}+1\right)}}R^{n}\,.$

Factorial in the complex plane

Amplitude and phase of factorial of complex argument

Representation through the gamma function allows evaluation of factorial of complex argument. Equilines of amplitude and phase of factorial are shown in figure. Let

$f=\rho e^{i\varphi }=(x+iy)!=\Gamma (x+iy+1)\,.$

Several levels of constant modulus (amplitude) Template:Mvar and constant phase Template:Mvar are shown. The grid covers the range Template:Math, Template:Math, with unit steps. The scratched line shows the level Template:Math.

Thin lines show intermediate levels of constant modulus and constant phase. At the poles at every negative integer, phase and amplitude are not defined. Equilines are dense in vicinity of singularities along negative integer values of the argument.

For Template:Math, the Taylor expansions can be used:

$z!=\sum _{n=0}^{\infty }g_{n}z^{n}\,.$

The first coefficients of this expansion are

Template:Mvar	Template:Mvar	Approximation
0	1	1
1	Template:Math	Template:Val
2	Template:Math	Template:Val
3	Template:Math	Template:Val

where Template:Mvar is the Euler–Mascheroni constant and Template:Math is the Riemann zeta function. Computer algebra systems such as SageMath can generate many terms of this expansion.

Approximations of the factorial

For the large values of the argument, the factorial can be approximated through the integral of the digamma function, using the continued fraction representation. This approach is due to T. J. Stieltjes (1894).Template:Citation needed Writing Template:Math where Template:Math is

$P(z)=p(z)+{\frac {\ln 2\pi }{2}}-z+\left(z+{\frac {1}{2}}\right)\ln(z)\,,$

Stieltjes gave a continued fraction for Template:Math:

$p(z)={\cfrac {a_{0}}{z+{\cfrac {a_{1}}{z+{\cfrac {a_{2}}{z+{\cfrac {a_{3}}{z+\ddots }}}}}}}}$

The first few coefficients Template:Math are

Template:Math	Template:Math
0	Template:Sfrac
1	Template:Sfrac
2	Template:Sfrac
3	Template:Sfrac
4	Template:Sfrac
5	Template:Sfrac
6	Template:Sfrac

There is a misconception that Template:Math or Template:Math for any complex Template:Math.Template:Citation needed Indeed, the relation through the logarithm is valid only for a specific range of values of Template:Mvar in the vicinity of the real axis, where Template:Math. The larger the real part of the argument, the smaller the imaginary part should be. However, the inverse relation, Template:Math, is valid for the whole complex plane apart from Template:Math. The convergence is poor in the vicinity of the negative part of the real axis;Template:Citation needed it is difficult to have good convergence of any approximation in the vicinity of the singularities. When Template:Math or Template:Math, the six coefficients above are sufficient for the evaluation of the factorial with complex double precision. For higher precision more coefficients can be computed by a rational QD scheme (Rutishauser's QD algorithm).

Non-extendability to negative integers

The relation Template:Math allows one to compute the factorial for an integer given the factorial for a smaller integer. The relation can be inverted so that one can compute the factorial for an integer given the factorial for a larger integer:

$(n-1)!={\frac {n!}{n}}.$

However, this recursion does not permit us to compute the factorial of a negative integer; use of the formula to compute (−1)! would require a division of a nonzero value by zero, and thus blocks us from computing a factorial value for every negative integer. Similarly, the gamma function is not defined for zero or negative integers, though it is defined for all other complex numbers.

Factorial-like products and functions

There are several other integer sequences similar to the factorial that are used in mathematics:

Double factorial

Template:Main The product of all the odd integers up to some odd positive integer Template:Mvar is called the double factorial of Template:Mvar, and denoted by Template:Math. That is,

$(2k-1)!!=\prod _{i=1}^{k}(2i-1)={\frac {(2k)!}{2^{k}k!}}={\frac {_{2k}P_{k}}{2^{k}}}={\frac {\left(2k\right)^{\underline {k}}}{2^{k}}}\,.$

For example, Template:Nowrap.

The sequence of double factorials for Template:Math starts as

1, 3, 15, 105, 945, Template:Val, Template:Val,... Template:OEIS

Double factorial notation may be used to simplify the expression of certain trigonometric integrals, to provide an expression for the values of the gamma function at half-integer arguments and the volume of hyperspheres, and to solve many counting problems in combinatorics including counting binary trees with labeled leaves and perfect matchings in complete graphs.

Multifactorials

A common related notation is to use multiple exclamation points to denote a multifactorial, the product of integers in steps of two (Template:Math), three (Template:Math), or more (see generalizations of the double factorial). The double factorial is the most commonly used variant, but one can similarly define the triple factorial (Template:Math) and so on. One can define the Template:Mvar-tuple factorial, denoted by Template:Math, recursively for positive integers as

$n!^{(k)}={\begin{cases}n&{\text{if }}0<n\leq k,\\n\left({(n-k)!}^{(k)}\right)&{\text{if }}n>k.\end{cases}}$

In addition, similarly to Template:Nowrap, one can define:

${n!}^{(k)}=1\quad {\text{if }}-k<n\leq 0$

For sufficiently large Template:Math, the ordinary single factorial function is expanded through the multifactorial functions as follows:

${\begin{aligned}n!&=n!!\cdot (n-1)!!\,,&n&\geq 1\\[5px]&=n!!!\cdot (n-1)!!!\cdot (n-2)!!!\,,&n&\geq 2\\[5px]&=\prod _{i=0}^{k-1}(n-i)!^{(k)},\quad {\text{ for }}k\in \mathbb {Z} ^{+}\,,&n&\geq k-1\,.\end{aligned}}$

In the same way that Template:Math is not defined for negative integers, and Template:Math is not defined for negative even integers, Template:Math is not defined for negative integers divisible by Template:Mvar.

Primorial

Template:Main The primorial of a natural number Template:Mvar Template:OEIS, denoted Template:Math, is similar to the factorial, but with the product taken only over the prime numbers less than or equal to Template:Mvar. That is,

$n\#=\prod _{p\leq n}p,$

where Template:Mvar ranges over the prime numbers less than or equal to Template:Mvar. For example, the primorial of 11 is

$11\#=11\times 7\times 5\times 3\times 2=2310.$

Superfactorial

Template:See alsoTemplate:Redirect Neil Sloane and Simon Plouffe defined a superfactorial in The Encyclopedia of Integer Sequences (Academic Press, 1995) to be the product of the first Template:Mvar factorials. So the superfactorial of 4 is

$\operatorname {sf} (4)=1!\times 2!\times 3!\times 4!=288\,.$

In general

${\begin{aligned}\operatorname {sf} (n)=\prod _{k=1}^{n}k!&=\prod _{k=1}^{n}k^{n-k+1}\\&=1^{n}\cdot 2^{n-1}\cdot 3^{n-2}\cdots (n-1)^{2}\cdot n^{1}\,.\end{aligned}}$

Equivalently, the superfactorial is given by the formula

$\operatorname {sf} (n)=\prod _{0\leq i<j\leq n}(j-i)$

which is the determinant of a Vandermonde matrix.

The sequence of superfactorials starts (from Template:Math) as

1, 1, 2, 12, 288, Template:Val, Template:Val, Template:Val,... Template:OEIS

By this definition, we can define the Template:Mvar-superfactorial of Template:Mvar (denoted Template:Math) as:

$\operatorname {sf} _{k}(n)={\begin{cases}n&{\text{if }}k=0\\\prod _{r=1}^{n}\operatorname {sf} _{k-1}(r)&{\text{if }}k\geq 1\end{cases}}$

The 2-superfactorials of Template:Mvar are

1, 1, 2, 24, Template:Val, Template:Val, Template:Val, Template:Val,... Template:OEIS

The 0-superfactorial of Template:Mvar is Template:Mvar.

Pickover’s superfactorial

In his 1995 book Keys to Infinity, Clifford Pickover defined a different function Template:Math that he called the superfactorial. It is defined by

$n\$\equiv {\begin{matrix}\underbrace {n!^{{n!}^{{\cdot }^{{\cdot }^{{\cdot }^{n!}}}}}} \\n!{\mbox{ copies of }}n!\end{matrix}}.$

This sequence of superfactorials starts

${\begin{aligned}1\$&=1\,,\\2\$&=2^{2}=4\,,\\3\$&=6^{6^{6^{6^{6^{6}}}}}\,.\end{aligned}}$

(Here, as is usual for compound exponentiation, the grouping is understood to be from right to left: Template:Math.)

This operation may also be expressed as the tetration

$n\$={}^{n!}(n!)\,,$

or using Knuth's up-arrow notation as

$n\$=(n!)\uparrow \uparrow (n!)\,.$

Hyperfactorial

Occasionally the hyperfactorial of n is considered. It is written as Template:Math and defined by

${\begin{aligned}H(n)&=\prod _{k=1}^{n}k^{k}\\&=1^{1}\cdot 2^{2}\cdot 3^{3}\cdots (n-1)^{n-1}\cdot n^{n}.\end{aligned}}$

For Template:Math the values of Template:Math are 1, 4, 108, Template:Val,... Template:OEIS.

The asymptotic growth rate is

$H(n)\sim An^{(6n^{2}+6n+1)/12}e^{-n^{2}/4}$

where Template:Mvar = 1.2824... is the Glaisher–Kinkelin constant. Template:Math ≈ Template:Val is already almost equal to a googol, and Template:Math ≈ Template:Val is almost of the same magnitude as the Shannon number, the theoretical number of possible chess games. Compared to the Pickover definition of the superfactorial, the hyperfactorial grows relatively slowly.

The hyperfactorial function can be generalized to complex numbers in a similar way as the factorial function. The resulting function is called the [[K-function|Template:Mvar-function]].

See also

• Alternating factorial
• Bhargava factorial
• Digamma function
• Exponential factorial
• Factorial number system
• Factorion
• List of factorial and binomial topics
• Pochhammer symbol, which gives the falling or rising factorial
• Subfactorial
• Trailing zeros of factorial
• Triangular number, the additive analogue of factorial

References

Sources

• Template:Citation
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• Template:Citation The publisher is given as "W.S." who may have been William Smith, possibly acting as agent for the Society of College Youths, to which society the "Dedicatory" is addressed.

Further reading

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External links

• Template:Britannica
• Template:Springer
• Template:MathWorld
• Template:PlanetMath