### 10.3 Combinatorial Functions

Operator: !!

  The double factorial operator.

  For an integer, float, or rational number `n`, `n!!` evaluates to the product `n (n-2) (n-4) (n-6) ... (n - 2 (k-1))` where `k` is equal to `entier (n/2)`, that is, the largest integer less than or equal to `n/2`. Note that this definition does not coincide with other published definitions for arguments which are not integers.

  For an even (or odd) integer `n`, `n!!` evaluates to the product of all the consecutive even (or odd) integers from 2 (or 1) through `n` inclusive.

  For an argument `n` which is not an integer, float, or rational, `n!!` yields a noun form `genfact (n, n/2, 2)`.

  ·

  Categories: Gamma and factorial functions ·Operators

Function: binomial (x, y)

  The binomial coefficient `x!/(y! (x - y)!)`. If x and y are integers, then the numerical value of the binomial coefficient is computed. If y, or x - y, is an integer, the binomial coefficient is expressed as a polynomial.

  Examples:

      (%i1) binomial (11, 7);
      (%o1)                          330

      (%i2) 11! / 7! / (11 - 7)!;
      (%o2)                          330

      (%i3) binomial (x, 7);
              (x - 6) (x - 5) (x - 4) (x - 3) (x - 2) (x - 1) x
      (%o3)   -------------------------------------------------
                                    5040

      (%i4) binomial (x + 7, x);
            (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (x + 6) (x + 7)
      (%o4) -------------------------------------------------------
                                     5040

      (%i5) binomial (11, y);
      (%o5)                    binomial(11, y)

  ·

  Categories: Number theory

Function: factcomb (expr)

  Tries to combine the coefficients of factorials in expr with the factorials themselves by converting, for example, `(n + 1)*n!` into `(n + 1)!`.

  `sumsplitfact` if set to `false` will cause `minfactorial` to be applied after a `factcomb`.

  Example:

      (%i1) sumsplitfact;
      (%o1)                         true

      (%i2) (n + 1)*(n + 1)*n!;
                                        2
      (%o2)                      (n + 1)  n!

      (%i3) factcomb (%);
      (%o3)                  (n + 2)! - (n + 1)!

      (%i4) sumsplitfact: not sumsplitfact;
      (%o4)                         false

      (%i5) (n + 1)*(n + 1)*n!;
                                        2
      (%o5)                      (n + 1)  n!

      (%i6) factcomb (%);
      (%o6)                 n (n + 1)! + (n + 1)!

  ·

  Categories: Gamma and factorial functions

Function: factorial
Operator: !

  Represents the factorial function. Maxima treats `factorial (x)` the same as `x!`.

  For any complex number `x`, except for negative integers, `x!` is defined as `gamma(x+1)`.

  For an integer `x`, `x!` simplifies to the product of the integers from 1 to `x` inclusive. `0!` simplifies to 1. For a real or complex number in float or bigfloat precision `x`, `x!` simplifies to the value of `gamma (x+1)`. For `x` equal to `n/2` where `n` is an odd integer, `x!` simplifies to a rational factor times `sqrt (%pi)` (since `gamma (1/2)` is equal to `sqrt (%pi)`).

  The option variables `factlim` and `gammalim` control the numerical evaluation of factorials for integer and rational arguments. The functions `minfactorial` and `factcomb` simplifies expressions containing factorials.

  The functions `gamma`, `bffac`, and `cbffac` are varieties of the gamma function. `bffac` and `cbffac` are called internally by `gamma` to evaluate the gamma function for real and complex numbers in bigfloat precision.

  `makegamma` substitutes `gamma` for factorials and related functions.

  Maxima knows the derivative of the factorial function and the limits for specific values like negative integers.

  The option variable `factorial_expand` controls the simplification of expressions like `(n+x)!`, where `n` is an integer.

  See also `binomial`.

  The factorial of an integer is simplified to an exact number unless the operand is greater than `factlim`. The factorial for real and complex numbers is evaluated in float or bigfloat precision.

      (%i1) factlim : 10;
      (%o1)                          10

      (%i2) [0!, (7/2)!, 8!, 20!];
                           105 sqrt(%pi)
      (%o2)            [1, -------------, 40320, 20!]
                                16

      (%i3) [4,77!, (1.0+%i)!];
      (%o3) [4, 77!, 0.3430658398165453 %i + 0.6529654964201667]

      (%i4) [2.86b0!, (1.0b0+%i)!];
      (%o4) [5.046635586910012b0, 3.430658398165454b-1 %i
                                                + 6.529654964201667b-1]

  The factorial of a known constant, or general expression is not simplified. Even so it may be possible to simplify the factorial after evaluating the operand.

      (%i1) [(%i + 1)!, %pi!, %e!, (cos(1) + sin(1))!];
      (%o1)      [(%i + 1)!, %pi!, %e!, (sin(1) + cos(1))!]

      (%i2) ev (%, numer, %enumer);
      (%o2) [0.3430658398165453 %i + 0.6529654964201667,
               7.188082728976031, 4.260820476357003, 1.227580202486819]

  Factorials are simplified, not evaluated. Thus `x!` may be replaced even in a quoted expression.

      (%i1) '([0!, (7/2)!, 4.77!, 8!, 20!]);
                105 sqrt(%pi)
      (%o1) [1, -------------, 81.44668037931197, 40320,
                     16
                                                   2432902008176640000]

  Maxima knows the derivative of the factorial function.

      (%i1) diff(x!,x);
      (%o1)                    x! psi (x + 1)
                                     0

  The option variable `factorial_expand` controls expansion and simplification of expressions with the factorial function.

      (%i1) (n+1)!/n!,factorial_expand:true;
      (%o1)                         n + 1

  ·

  Categories: Gamma and factorial functions ·Operators

Option variable: factlim

  Default value: 100000

  `factlim` specifies the highest factorial which is automatically expanded. If it is -1 then all integers are expanded.

  ·

  Categories: Gamma and factorial functions

Option variable: factorial_expand

  Default value: false

  The option variable `factorial_expand` controls the simplification of expressions like `(n+1)!`, where `n` is an integer. See `factorial` for an example.

  ·

  Categories: Gamma and factorial functions

Function: genfact (x, y, z)

  Returns the generalized factorial, defined as `x (x-z) (x - 2 z) ... (x - (y - 1) z)`. Thus, when x is an integer, `genfact (x, x, 1) = x!` and `genfact (x, x/2, 2) = x!!`.

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  Categories: Gamma and factorial functions

Function: minfactorial (expr)

  Examines expr for occurrences of two factorials which differ by an integer. `minfactorial` then turns one into a polynomial times the other.

      (%i1) n!/(n+2)!;
                                     n!
      (%o1)                       --------
                                  (n + 2)!
      (%i2) minfactorial (%);
                                      1
      (%o2)                    ---------------
                               (n + 1) (n + 2)

  ·

  Categories: Number theory

Option variable: sumsplitfact

  Default value: `true`

  When `sumsplitfact` is `false`, `minfactorial` is applied after a `factcomb`.

      (%i1) sumsplitfact;
      (%o1)                         true

      (%i2) n!/(n+2)!;
                                     n!
      (%o2)                       --------
                                  (n + 2)!

      (%i3) factcomb(%);
                                     n!
      (%o3)                       --------
                                  (n + 2)!

      (%i4) sumsplitfact: not sumsplitfact ;
      (%o4)                         false

      (%i5) n!/(n+2)!;
                                     n!
      (%o5)                       --------
                                  (n + 2)!

      (%i6) factcomb(%);
                                      1
      (%o6)                    ---------------
                               (n + 1) (n + 2)

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  Categories: Gamma and factorial functions ·Simplification flags and variables