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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!=5×4×3×2×1=120.{displaystyle 5!=5times 4times 3times 2times 1=120,.}

The value of 0! is 1, according to the convention for an empty product.^[1]

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.^[2] In 1677, Fabian Stedman described factorials as applied to change ringing, a musical art involving the ringing of many tuned bells.^[3] 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.^[4]

The notation n! was introduced by the French mathematician Christian Kramp in 1808.^[5]

Definition

The factorial function is defined by the product

n!=1⋅2⋅3⋅…⋅(n−2)⋅(n−1)⋅n,{displaystyle n!=1cdot 2cdot 3cdot 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!=∏i=1ni.{displaystyle n!=prod _{i=1}^{n}i.}

From these formulas, one may derive the recurrence relation

n!=n⋅(n−1)!.{displaystyle n!=ncdot (n-1)!.}

For example, one has

5!=5⋅4!6!=6⋅5!50!=50⋅49!{displaystyle {begin{aligned}5!&=5cdot 4!\6!&=6cdot 5!\50!&=50cdot 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{displaystyle 0!=1}

so that

1!=1⋅0!=1.{displaystyle 1!=1cdot 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
  

(00)=0!0!0!=1{displaystyle {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

(nn)=n!n!0!=1{displaystyle {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:

ex=∑n=0∞xnn!.{displaystyle 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.^[6][7]
  


• 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)⋯(n−(k−1))=n!(n−k)!=nk_{displaystyle (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

n(n−1)(n−2)⋯(n−k+1)k(k−1)(k−2)⋯1=nk_k!=n!(n−k)!k!=(nk).{displaystyle {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^[8] as the binomial coefficient, because it is also the coefficient of x^k in (1 + x)ⁿ. The term nk_{displaystyle 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,^[9] 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.^[10]
  


• Factorials can be useful to facilitate expression manipulation. For instance the number of k-permutations of n can be written as

nk_=n!(n−k)!;{displaystyle 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^[7][8] of binomial coefficients:

(nk)=nk_k!=n!(n−k)!k!=nn−k_(n−k)!=(nn−k).{displaystyle {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!=Dnxn=dndxnxn{displaystyle n!=D^{n},x^{n}={frac {d^{n}}{dx^{n}}},x^{n}}

where Dⁿxⁿ is Euler's notation for the nth derivative of xⁿ.^[11]

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!=∑x=1nln⁡x.{displaystyle 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:

∫1nln⁡xdx≤∑x=1nln⁡x≤∫0nln⁡(x+1)dx{displaystyle int _{1}^{n}ln x,dxleq sum _{x=1}^{n}ln xleq int _{0}^{n}ln(x+1),dx}

which gives us the estimate

nln⁡(ne)+1≤ln⁡n!≤(n+1)ln⁡(n+1e)+1.{displaystyle nln left({frac {n}{e}}right)+1leq 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

(ne)ne≤n!≤(n+1e)n+1e.{displaystyle left({frac {n}{e}}right)^{n}eleq 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!∼2πn(ne)n.{displaystyle n!sim {sqrt {2pi 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:

2πn(ne)n<n!<2πn(ne)ne112n.{displaystyle {sqrt {2pi n}}left({frac {n}{e}}right)^{n}<n!<{sqrt {2pi n}}left({frac {n}{e}}right)^{n}e^{frac {1}{12n}},.}

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

ln⁡n!≈nln⁡n−n+ln⁡(n(1+4n(1+2n)))6+ln⁡π2⟹n!≈2πn(ne)n(1+12n+18n2)16.{displaystyle {begin{aligned}ln n!&approx nln n-n+{frac {ln {Bigl (}n{bigl (}1+4n(1+2n){bigr )}{Bigr )}}{6}}+{frac {ln pi }{2}}\[6px]Longrightarrow ;n!&approx {sqrt {2pi 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!≈2πn(ne)nexp⁡(112n−1360n3).{displaystyle n!approx {sqrt {2pi 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.^[12]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.^[13]

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).^[14] Peter Luschny presents source code and benchmarks for several efficient factorial algorithms, with or without the use of a prime sieve.^[15]

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)!≡0(modn).{displaystyle (n-1)!equiv 0{pmod {n}}.}

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

(p−1)!≡−1(modp){displaystyle (p-1)!equiv -1{pmod {p}}}

if and only if p is prime.^[16][17]

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

∑i=1∞⌊npi⌋{displaystyle sum _{i=1}^{infty }leftlfloor {frac {n}{p^{i}}}rightrfloor }

or, equivalently,

n−sp(n)p−1,{displaystyle {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.^[18] 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:

∑n=0∞1n!=11+11+12+16+124+1120+⋯=e.{displaystyle 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:

∑n=0∞1(n+2)n!=12+13+18+130+1144+⋯=1.{displaystyle 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.^[19]

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.

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

Γ(z)=∫0∞tz−1e−tdt.{displaystyle Gamma (z)=int _{0}^{infty }t^{z-1}e^{-t},dt,.}

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

n!=Γ(n+1).{displaystyle n!=Gamma (n+1),.}

Euler's original formula for the gamma function was

Γ(z)=limn→∞nzn!∏k=0n(z+k).{displaystyle Gamma (z)=lim _{nto 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

Π(z)=∫0∞tze−tdt.{displaystyle Pi (z)=int _{0}^{infty }t^{z}e^{-t},dt,.}

In terms of the gamma function, the pi function is

Π(z)=Γ(z+1).{displaystyle 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!=Π(n) for n∈N.{displaystyle n!=Pi (n)quad {text{ for }}nin 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

Π(z)=zΠ(z−1).{displaystyle Pi (z)=zPi (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

Γ(n+1)=nΓ(n).{displaystyle Gamma (n+1)=nGamma (n),.}

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

z!=Π(z){displaystyle z!=Pi (z)}

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

Γ(12)=(−12)!=Π(−12)=π,{displaystyle 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,

Γ(12+n)=(−12+n)!=Π(−12+n)=π∏k=1n2k−12=(2n)!4nn!π=(2n−1)!22n−1(n−1)!π.{displaystyle {begin{aligned}&Gamma left({frac {1}{2}}+nright)=left(-{frac {1}{2}}+nright)!=Pi left(-{frac {1}{2}}+nright)\[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,

Γ(92)=72!=Π(72)=12⋅32⋅52⋅72π=8!444!π=7!273!π=10516π≈11.631728…{displaystyle 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 11.631,728ldots }

It also follows that for n ∈ N,

Γ(12−n)=(−12−n)!=Π(−12−n)=π∏k=1n21−2k=(−4)nn!(2n)!π.{displaystyle Gamma left({frac {1}{2}}-nright)=left(-{frac {1}{2}}-nright)!=Pi left(-{frac {1}{2}}-nright)={sqrt {pi }}prod _{k=1}^{n}{frac {2}{1-2k}}={frac {left(-4right)^{n}n!}{(2n)!}}{sqrt {pi }},.}

For example,

Γ(−52)=(−72)!=Π(−72)=2−1⋅2−3⋅2−5π=(−4)33!6!π=−815π≈−0.945308…{displaystyle 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(-4right)^{3}3!}{6!}}{sqrt {pi }}=-{frac {8}{15}}{sqrt {pi }}approx -0.945,308ldots }

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.^[20]

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:

n!=Π(n)=∏k=1∞(k+1k)nkn+k=[(21)n1n+1][(32)n2n+2][(43)n3n+3]⋯{displaystyle {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

Vn=πn2Γ(n2+1)Rn.{displaystyle V_{n}={frac {pi ^{frac {n}{2}}}{Gamma left({frac {n}{2}}+1right)}}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=ρeiφ=(x+iy)!=Γ(x+iy+1).{displaystyle f=rho e^{ivarphi }=(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!=∑n=0∞gnzn.{displaystyle z!=sum _{n=0}^{infty }g_{n}z^{n},.}

The first coefficients of this expansion are

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).^{[citation needed]} Writing z! = e^P(z) where P(z) is

P(z)=p(z)+ln⁡2π2−z+(z+12)ln⁡(z),{displaystyle P(z)=p(z)+{frac {ln 2pi }{2}}-z+left(z+{frac {1}{2}}right)ln(z),,}

Stieltjes gave a continued fraction for p(z):

p(z)=a0z+a1z+a2z+a3z+⋱{displaystyle 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^[21]

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.^{[citation needed]} 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;^{[citation needed]} 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).^[22]

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)!=n!n.{displaystyle (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

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!!.^[23] That is,

(2k−1)!!=∏i=1k(2i−1)=(2k)!2kk!=2kPk2k=(2k)k_2k.{displaystyle (2k-1)!!=prod _{i=1}^{k}(2i-1)={frac {(2k)!}{2^{k}k!}}={frac {_{2k}P_{k}}{2^{k}}}={frac {left(2kright)^{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,^[24] to provide an expression for the values of the gamma function at half-integer arguments and the volume of hyperspheres,^[25] and to solve many counting problems in combinatorics including counting binary trees with labeled leaves and perfect matchings in complete graphs.^[23][26]

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)={nif 0<n≤kn((n−k)!(k))if n>k.{displaystyle n!^{(k)}={begin{cases}n&{text{if }}0<nleq k\nleft({(n-k)!}^{(k)}right)&{text{if }}n>kend{cases}}.}

In addition, similarly to 0! = 1!/1 = 1, one can define:

n!(k)=1if −k<n≤0{displaystyle {n!}^{(k)}=1quad {text{if }}-k<nleq 0}

For sufficiently large n ≥ 1, the ordinary single factorial function is expanded through the multifactorial functions as follows:

n!=n!!⋅(n−1)!!,n≥1=n!!!⋅(n−1)!!!⋅(n−2)!!!,n≥2=∏i=0k−1(n−i)!(k), for k∈Z+,n≥k−1.{displaystyle {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 }}kin 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

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

sf⁡(4)=1!×2!×3!×4!=288.{displaystyle operatorname {sf} (4)=1!times 2!times 3!times 4!=288,.}

In general

sf⁡(n)=∏k=1nk!=∏k=1nkn−k+1=1n⋅2n−1⋅3n−2⋯(n−1)2⋅n1.{displaystyle {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

sf⁡(n)=∏0≤i<j≤n(j−i){displaystyle operatorname {sf} (n)=prod _{0leq i<jleq 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:

sfk⁡(n)={nif k=0∏r=1nsfk−1⁡(r)if k≥1{displaystyle operatorname {sf} _{k}(n)={begin{cases}n&{text{if }}k=0\prod _{r=1}^{n}operatorname {sf} _{k-1}(r)&{text{if }}kgeq 1end{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$≡n!n!⋅⋅⋅n!⏟n! copies of n!.{displaystyle n$equiv {begin{matrix}underbrace {n!^{{n!}^{{cdot }^{{cdot }^{{cdot }^{n!}}}}}} \n!{mbox{ copies of }}n!end{matrix}}.}

This sequence of superfactorials starts

1$=1,2$=22=4,3$=666666.{displaystyle {begin{aligned}1$&=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$=n!(n!),{displaystyle n$={}^{n!}(n!),,}

or using Knuth's up-arrow notation as

n$=(n!)↑↑(n!).{displaystyle n$=(n!)uparrow uparrow (n!),.}

Hyperfactorial

Occasionally the hyperfactorial of n is considered. It is written as H(n) and defined by

H(n)=∏k=1nkk=11⋅22⋅33⋯(n−1)n−1⋅nn.{displaystyle {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)∼An6n2+6n+112e−n24{displaystyle H(n)sim An^{frac {6n^{2}+6n+1}{12}}e^{-{frac {n^{2}}{4}}}}

where A = 1.2824... is the Glaisher–Kinkelin constant.^[27]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

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

External links

Wikimedia Commons has media related to Factorial (function).

• 
  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.