Factorial

Selected members of the factorial sequence (sequence A000142 in the OEIS); values specified in scientific notation are rounded to the displayed precision

n	n!
0	1
1	1
2	2
3	6
4	24
5	120
6	720
7	7003504000000000000♠5040
8	7004403200000000000♠40320
9	7005362880000000000♠362880
10	7006362880000000000♠3628800
11	7007399168000000000♠39916800
12	7008479001600000000♠479001600
13	7009622702080000000♠6227020800
14	7010871782912000000♠87178291200
15	7012130767436800000♠1307674368000
16	7013209227898880000♠20922789888000
17	7014355687428096000♠355687428096000
18	7015640237370572800♠6402373705728000
19	7017121645100408832♠121645100408832000
20	7018243290200817664♠2432902008176640000
25	7025155112100400000♠1.551121004×1025
50	7064304140932000000♠3.041409320×1064
70	7100119785716700000♠1.197857167×10100
100	7157933262154400000♠9.332621544×10157
450	9000000000000000000♠1.733368733×101000
7003100000000000000♠1000	9000000000000000000♠4.023872601×102567
7003324900000000000♠3249	9000000000000000000♠6.412337688×1010000
7004100000000000000♠10000	9000000000000000000♠2.846259681×1035659
7004252060000000000♠25206	9000000000000000000♠1.205703438×10100000
7005100000000000000♠100000	9000000000000000000♠2.824229408×10456573
7005205023000000000♠205023	9000000000000000000♠2.503898932×101000004
7006100000000000000♠1000000	9000000000000000000♠8.263931688×105565708
7100100000000000000♠10100	107101995657055180894♠10101.9981097754820

In mathematics, the factorial of a positive integer n, denoted by n!, is the product of all positive integers less than or equal to n. 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.

The factorial operation is encountered in many areas of mathematics, notably in combinatorics, algebra, and mathematical analysis. Its most basic occurrence is the fact that there are n! ways to arrange n distinct objects into a sequence. These arrangements are called the permutations of the set of objects.

The definition of the factorial function can also be extended to non-integer arguments, while retaining its most important properties; this involves more advanced mathematics, notably techniques from mathematical analysis.

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. After describing a recursive approach, Stedman gives a statement of a factorial (using the language of the original):

Now the nature of these methods is such, that the changes on one number comprehends [includes] the changes on all lesser numbers ... insomuch that a compleat Peal of changes on one number seemeth to be formed by uniting of the compleat Peals on all lesser numbers into one entire body.

The notation n! was introduced by the French mathematician Christian Kramp in 1808.

Definition

The factorial function is defined by the product

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

initially for integer n ≥ 1. This may be written in the 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

In order for the recurrence relation to be extended to n = 0, it is necessary to define

$0!=1$

so that

$1!=1\cdot 0!=1.$

There are several independent reasons to view this definition as harmonious. These include the following:

• In the case n = 0, the definition of n! 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 n elements among a set of n 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:

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

Factorial of a non-integer

The factorial function can also be defined for non-integer values using more advanced mathematics (the gamma function n! = Γ(n + 1)), detailed in the section below. This more generalized definition is used by advanced calculators and mathematical software such as Maple, Mathematica, or APL.

Applications

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

• There are n! different ways of arranging n 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 k-combinations (subsets of k elements) from a set with n elements. One can obtain such a combination by choosing a k-permutation: successively selecting and removing one element of the set, k times, for a total of

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

possibilities. This however produces the k-combinations in a particular order that one wishes to ignore; since each k-combination is obtained in k! different ways, the correct number of k-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 x^k in (1 + x)ⁿ. 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 nth derivative of xⁿ being equivalent to n!.
• Factorials are also used extensively in probability theory.
• Factorials can be useful to facilitate expression manipulation. For instance the number of k-permutations of n 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 Dⁿ xⁿ is Euler's notation for the nth derivative of xⁿ.

Rate of growth and approximations for large n

Plot of the natural logarithm of the factorial

As n grows, the factorial n! increases faster than all polynomials and exponential functions (but slower than double exponential functions) in n.

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

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

The graph of the function f(n) = ln n! is shown in the figure on the right. It looks approximately linear for all reasonable values of n, but this intuition is false. We get one of the simplest approximations for ln n! 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 ln n! ∼ n ln n (see Big O notation). This result plays a key role in the analysis of the computational complexity of sorting algorithms (see comparison sort). From the bounds on ln n! 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 n we have (n/3)ⁿ < n!, and for all n ≥ 6 we have n! < (n/2)ⁿ.

Comparison of Stirling's approximation with the factorial

For large n we get a better estimate for the number n! 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 n factorial lies between this and the next approximation:

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

Another approximation for ln n! is given by Srinivasa Ramanujan (Ramanujan 1988)

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

Both this and the previous approximation give a relative error on the order of 1/n³, but Ramanujan's is about four times more accurate. However, if we use two correction terms (as in Ramanujan's approximation) the relative error will be of order 1/n⁵:

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

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 n (if any) will compute n!, 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 69! < 10¹⁰⁰ < 70!. 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 n! for values of n up to 7005249999000000000♠249999, and up to 7007200000000000000♠20000000! 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 ((1 × 2) × 3) × 4..., 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 n! from its prime factorization. As documented by Peter Borwein, prime factorization allows n! to be computed in time O(n(log n log log n)²), 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, n! is necessarily divisible by all prime numbers up to and including n. As a consequence, n > 5 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 p is prime.

Legendre's formula gives the multiplicity of the prime p occurring in the prime factorization of n! as

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

or, equivalently,

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

where s_p(n) denotes the sum of the standard base-p digits of n.

Adding 1 to a factorial n! yields a number that is divisible by a prime larger than n. This fact can be used to prove Euclid's theorem that the number of primes is infinite. Primes of the form n! ± 1 are called factorial primes.

Series of reciprocals

The reciprocals of factorials produce a convergent series whose sum is the exponential base e:

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

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.

Extension of factorial to non-integer values of argument

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.

Main article: Gamma function

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 Γ(z). It is defined for all complex numbers z except for the non-positive integers, and given when the real part of z is positive by

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

Its relation to the factorial is that for any natural number n

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

Euler's original formula for the gamma function was

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

Another function that also "fills in" the values of the factorial (but with no shift in the argument), originally introduced by Carl Friedrich Gauss, that is sometimes used, is called the pi function, denoted Π(z) for real numbers z ≥ 0. It is defined by

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

In terms of the gamma function, the pi function is

$\Pi (z)=\Gamma (z+1)\,.$

The factorial function, generalized to all real numbers except negative integers. For example, 0! = 1! = 1, (−1/2)! = √π, 1/2! = √π/2.

The pi function properly extends the factorial in that

$n!=\Pi (n)\quad {\text{ for }}n\in \mathbf {N} \,.$

In addition to this, the pi function satisfies the same recurrence as factorials do, but at every complex value z 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

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

Since the factorial is extended by the pi function, for every complex value z where it is defined, we can write:

$z!=\Pi (z)$

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 n ∈ N,

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

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 n ∈ N,

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

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 Γ(n + 1) = nΓ(n), 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 Π(n) = nΠ(n − 1) 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 (Hadamard 1894) 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 n-dimensional hypersphere of radius R is

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

Factorial at 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

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

Several levels of constant modulus (amplitude) ρ and constant phase φ are shown. The grid covers the range −3 ≤ x ≤ 3, −2 ≤ y ≤ 2, with unit steps. The scratched line shows the level φ = ±π.

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 |z| < 1, the Taylor expansions can be used:

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

The first coefficients of this expansion are

n	gn	approximation
0	1	1
1	−γ	3000422784335100000♠−0.5772156649
2	π2/12 + γ2/2	6999989055995500000♠0.9890559955
3	−ζ(3)/3 − π2/12 − γ3/6	3000092520924000000♠−0.9074790760

where γ is the Euler–Mascheroni constant and ζ(z) 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). Writing z! = e^P(z) where P(z) 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 p(z):

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

The first few coefficients a_n are

n	an
0	1/12
1	1/30
2	53/210
3	195/371
4	7004229990000000000♠22999/7004227370000000000♠22737
5	7007299445230000000♠29944523/7007197331420000000♠19733142
6	7011109535241009000♠109535241009/7010482642754620000♠48264275462

There is a misconception that ln z! = P(z) or ln Γ(z + 1) = P(z) for any complex z ≠ 0. Indeed, the relation through the logarithm is valid only for a specific range of values of z in the vicinity of the real axis, where −π < Im(Γ(z + 1)) < π. The larger the real part of the argument, the smaller the imaginary part should be. However, the inverse relation, z! = e^P(z), is valid for the whole complex plane apart from z = 0. The convergence is poor in the vicinity of the negative part of the real axis; it is difficult to have good convergence of any approximation in the vicinity of the singularities. When |Im z| > 2 or Re z > 2, 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 n! = n × (n − 1)! 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:

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

Note, however, that 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 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

Main article: Double factorial

The product of all the odd integers up to some odd positive integer n is called the double factorial of n, and denoted by n!!. 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, 9!! = 1 × 3 × 5 × 7 × 9 = 945.

The sequence of double factorials for n = 1, 3, 5, 7,... starts as

1, 3, 15, 105, 945, 7004103950000000000♠10395, 7005135135000000000♠135135,... (sequence A001147 in the 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 (n!!), three (n!!!), 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 (n!!!) and so on. One can define the k-tuple factorial, denoted by n!^(k), 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 0! = 1!/1 = 1, one can define:

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

For sufficiently large n ≥ 1, 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 n! is not defined for negative integers, and n!! is not defined for negative even integers, n!^(k) is not defined for negative integers divisible by k.

Primorial

The primorial (sequence A002110 in the OEIS) is similar to the factorial, but with the product taken only over the prime numbers.

Superfactorial

Main article: Large numbers

"N$" redirects here. For the currency, see Namibian dollar.

Neil Sloane and Simon Plouffe defined a superfactorial in The Encyclopedia of Integer Sequences (Academic Press, 1995) to be the product of the first n 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 n = 0) as

1, 1, 2, 12, 288, 7004345600000000000♠34560, 7007248832000000000♠24883200, 7011125411328000000♠125411328000,... (sequence A000178 in the OEIS)

By this definition, we can define the k-superfactorial of n (denoted sf_k(n)) 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 n are

1, 1, 2, 24, 7003691200000000000♠6912, 7008238878720000000♠238878720, 7015594406696550400♠5944066965504000, 7026745453331864786♠745453331864786829312000000,... (sequence A055462 in the OEIS)

The 0-superfactorial of n is n.

Pickover’s superfactorial

In his 1995 book Keys to Infinity, Clifford Pickover defined a different function n$ 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: a^bc = a^(bc).)

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 H(n) 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 n = 1, 2, 3, 4,... the values of H(n) are 1, 4, 108, 7004276480000000000♠27648,... (sequence A002109 in the OEIS).

The asymptotic growth rate is

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

where A = 1.2824... is the Glaisher–Kinkelin constant. H(14) ≈ 7099184740000000000♠1.8474×10⁹⁹ is already almost equal to a googol, and H(15) ≈ 7116808960000000000♠8.0896×10¹¹⁶ 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.

See also

• Alternating factorial
• Bhargava factorial
• Digamma function
• Exponential factorial
• Factorial number system
• Factorial prime
• 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

• Bostock, Linda; Chandler, Suzanne; Rourke, C. (2014-11-01), Further Pure Mathematics, Nelson Thornes, ISBN 9780859501033
• Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1988), Concrete Mathematics, Reading, MA: Addison-Wesley, ISBN 0-201-14236-8
• Guy, Richard K. (2004), "E24 Irrationality sequences", Unsolved problems in number theory (3rd ed.), Springer-Verlag, ISBN 0-387-20860-7, Zbl 1058.11001
• Higgins, Peter (2008), Number Story: From Counting to Cryptography, New York: Copernicus, ISBN 978-1-84800-000-1
• Stedman, Fabian (1677), Campanalogia, London 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

• Hadamard, M. J. (1894), "Sur L'Expression Du Produit 1·2·3· · · · ·(n−1) Par Une Fonction Entière" (PDF), Œuvres de Jacques Hadamard (in French), Paris (1968): Centre National de la Recherche Scientifiques
• Ramanujan, Srinivasa (1988), The Lost Notebook and Other Unpublished Papers, Springer Berlin, p. 339, ISBN 3-540-18726-X

External links

• Hazewinkel, Michiel, ed. (2001) [1994], "Factorial", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
• Weisstein, Eric W. "Factorial". MathWorld.
• Factorial at PlanetMath.org.