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<a name="Combinatorial-Functions"></a>
<div class="header">
<p>
Next: <a href="maxima_51.html#Root-Exponential-and-Logarithmic-Functions" accesskey="n" rel="next">Root Exponential and Logarithmic Functions</a>, Previous: <a href="maxima_49.html#Functions-for-Complex-Numbers" accesskey="p" rel="previous">Functions for Complex Numbers</a>, Up: <a href="maxima_47.html#Elementary-Functions" accesskey="u" rel="up">Elementary Functions</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
</div>
<a name="Combinatorial-Functions-1"></a>
<h3 class="section">10.3 Combinatorial Functions</h3>
<a name="g_t_0021_0021"></a><a name="Item_003a-MathFunctions_002fdeffn_002f_0021_0021"></a><dl>
<dt><a name="index-_0021_0021"></a>Operator: <strong>!!</strong></dt>
<dd><a name="index-Double-factorial"></a>
<p>The double factorial operator.
</p>
<p>For an integer, float, or rational number <code>n</code>, <code>n!!</code> evaluates to the
product <code>n (n-2) (n-4) (n-6) ... (n - 2 (k-1))</code> where <code>k</code> is equal to
<code>entier (n/2)</code>, that is, the largest integer less than or equal to
<code>n/2</code>. Note that this definition does not coincide with other published
definitions for arguments which are not integers.
</p>
<p>For an even (or odd) integer <code>n</code>, <code>n!!</code> evaluates to the product of
all the consecutive even (or odd) integers from 2 (or 1) through <code>n</code>
inclusive.
</p>
<p>For an argument <code>n</code> which is not an integer, float, or rational, <code>n!!</code>
yields a noun form <code>genfact (n, n/2, 2)</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Gamma-and-factorial-functions">Gamma and factorial functions</a>
·<a href="maxima_424.html#Category_003a-Operators">Operators</a>
·</div></dd></dl>
<a name="binomial"></a><a name="Item_003a-MathFunctions_002fdeffn_002fbinomial"></a><dl>
<dt><a name="index-binomial"></a>Function: <strong>binomial</strong> <em>(<var>x</var>, <var>y</var>)</em></dt>
<dd>
<p>The binomial coefficient <code><var>x</var>!/(<var>y</var>! (<var>x</var> - <var>y</var>)!)</code>.
If <var>x</var> and <var>y</var> are integers, then the numerical value of the binomial
coefficient is computed. If <var>y</var>, or <var>x - y</var>, is an integer, the
binomial coefficient is expressed as a polynomial.
</p>
<p>Examples:
</p>
<div class="example">
<pre class="example">(%i1) binomial (11, 7);
(%o1) 330
</pre><pre class="example">(%i2) 11! / 7! / (11 - 7)!;
(%o2) 330
</pre><pre class="example">(%i3) binomial (x, 7);
(x - 6) (x - 5) (x - 4) (x - 3) (x - 2) (x - 1) x
(%o3) -------------------------------------------------
5040
</pre><pre class="example">(%i4) binomial (x + 7, x);
(x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (x + 6) (x + 7)
(%o4) -------------------------------------------------------
5040
</pre><pre class="example">(%i5) binomial (11, y);
(%o5) binomial(11, y)
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Number-theory">Number theory</a>
·</div></dd></dl>
<a name="factcomb"></a><a name="Item_003a-MathFunctions_002fdeffn_002ffactcomb"></a><dl>
<dt><a name="index-factcomb"></a>Function: <strong>factcomb</strong> <em>(<var>expr</var>)</em></dt>
<dd>
<p>Tries to combine the coefficients of factorials in <var>expr</var>
with the factorials themselves by converting, for example, <code>(n + 1)*n!</code>
into <code>(n + 1)!</code>.
</p>
<p><code><a href="#sumsplitfact">sumsplitfact</a></code> if set to <code>false</code> will cause <code><a href="#minfactorial">minfactorial</a></code> to be
applied after a <code>factcomb</code>.
</p>
<p>Example:
</p>
<div class="example">
<pre class="example">(%i1) sumsplitfact;
(%o1) true
</pre><pre class="example">(%i2) (n + 1)*(n + 1)*n!;
2
(%o2) (n + 1) n!
</pre><pre class="example">(%i3) factcomb (%);
(%o3) (n + 2)! - (n + 1)!
</pre><pre class="example">(%i4) sumsplitfact: not sumsplitfact;
(%o4) false
</pre><pre class="example">(%i5) (n + 1)*(n + 1)*n!;
2
(%o5) (n + 1) n!
</pre><pre class="example">(%i6) factcomb (%);
(%o6) n (n + 1)! + (n + 1)!
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Gamma-and-factorial-functions">Gamma and factorial functions</a>
·</div></dd></dl>
<a name="g_t_0021"></a><a name="factorial"></a><a name="Item_003a-MathFunctions_002fdeffn_002ffactorial"></a><dl>
<dt><a name="index-factorial"></a>Function: <strong>factorial</strong></dt>
<dd><a name="Item_003a-MathFunctions_002fdeffn_002f_0021"></a></dd><dt><a name="index-_0021"></a>Operator: <strong>!</strong></dt>
<dd>
<p>Represents the factorial function. Maxima treats <code>factorial (<var>x</var>)</code>
the same as <code><var>x</var>!</code>.
</p>
<p>For any complex number <code>x</code>, except for negative integers, <code>x!</code> is
defined as <code>gamma(x+1)</code>.
</p>
<p>For an integer <code>x</code>, <code>x!</code> simplifies to the product of the integers
from 1 to <code>x</code> inclusive. <code>0!</code> simplifies to 1. For a real or complex
number in float or bigfloat precision <code>x</code>, <code>x!</code> simplifies to the
value of <code>gamma (x+1)</code>. For <code>x</code> equal to <code>n/2</code> where <code>n</code> is
an odd integer, <code>x!</code> simplifies to a rational factor times
<code>sqrt (%pi)</code> (since <code>gamma (1/2)</code> is equal to <code>sqrt (%pi)</code>).
</p>
<p>The option variables <code><a href="#factlim">factlim</a></code> and <code><a href="maxima_87.html#gammalim">gammalim</a></code> control the numerical
evaluation of factorials for integer and rational arguments. The functions
<code><a href="#minfactorial">minfactorial</a></code> and <code><a href="#factcomb">factcomb</a></code> simplifies expressions containing
factorials.
</p>
<p>The functions <code><a href="maxima_87.html#gamma">gamma</a></code>, <code><a href="maxima_87.html#bffac">bffac</a></code>, and <code><a href="maxima_87.html#cbffac">cbffac</a></code> are
varieties of the gamma function. <code>bffac</code> and <code>cbffac</code> are called
internally by <code>gamma</code> to evaluate the gamma function for real and complex
numbers in bigfloat precision.
</p>
<p><code><a href="maxima_87.html#makegamma">makegamma</a></code> substitutes <code>gamma</code> for factorials and related functions.
</p>
<p>Maxima knows the derivative of the factorial function and the limits for
specific values like negative integers.
</p>
<p>The option variable <code><a href="#factorial_005fexpand">factorial_expand</a></code> controls the simplification of
expressions like <code>(n+x)!</code>, where <code>n</code> is an integer.
</p>
<p>See also <code><a href="#binomial">binomial</a></code>.
</p>
<p>The factorial of an integer is simplified to an exact number unless the operand
is greater than <code>factlim</code>. The factorial for real and complex numbers is
evaluated in float or bigfloat precision.
</p>
<div class="example">
<pre class="example">(%i1) factlim : 10;
(%o1) 10
</pre><pre class="example">(%i2) [0!, (7/2)!, 8!, 20!];
105 sqrt(%pi)
(%o2) [1, -------------, 40320, 20!]
16
</pre><pre class="example">(%i3) [4,77!, (1.0+%i)!];
(%o3) [4, 77!, 0.3430658398165453 %i + 0.6529654964201667]
</pre><pre class="example">(%i4) [2.86b0!, (1.0b0+%i)!];
(%o4) [5.046635586910012b0, 3.430658398165454b-1 %i
+ 6.529654964201667b-1]
</pre></div>
<p>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.
</p>
<div class="example">
<pre class="example">(%i1) [(%i + 1)!, %pi!, %e!, (cos(1) + sin(1))!];
(%o1) [(%i + 1)!, %pi!, %e!, (sin(1) + cos(1))!]
</pre><pre class="example">(%i2) ev (%, numer, %enumer);
(%o2) [0.3430658398165453 %i + 0.6529654964201667,
7.188082728976031, 4.260820476357003, 1.227580202486819]
</pre></div>
<p>Factorials are simplified, not evaluated.
Thus <code>x!</code> may be replaced even in a quoted expression.
</p>
<div class="example">
<pre class="example">(%i1) '([0!, (7/2)!, 4.77!, 8!, 20!]);
105 sqrt(%pi)
(%o1) [1, -------------, 81.44668037931197, 40320,
16
2432902008176640000]
</pre></div>
<p>Maxima knows the derivative of the factorial function.
</p>
<div class="example">
<pre class="example">(%i1) diff(x!,x);
(%o1) x! psi (x + 1)
0
</pre></div>
<p>The option variable <code>factorial_expand</code> controls expansion and
simplification of expressions with the factorial function.
</p>
<div class="example">
<pre class="example">(%i1) (n+1)!/n!,factorial_expand:true;
(%o1) n + 1
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Gamma-and-factorial-functions">Gamma and factorial functions</a>
·<a href="maxima_424.html#Category_003a-Operators">Operators</a>
·</div></dd></dl>
<a name="factlim"></a><a name="Item_003a-MathFunctions_002fdefvr_002ffactlim"></a><dl>
<dt><a name="index-factlim"></a>Option variable: <strong>factlim</strong></dt>
<dd><p>Default value: 100000
</p>
<p><code>factlim</code> specifies the highest factorial which is
automatically expanded. If it is -1 then all integers are expanded.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Gamma-and-factorial-functions">Gamma and factorial functions</a>
·</div></dd></dl>
<a name="factorial_005fexpand"></a><a name="Item_003a-MathFunctions_002fdefvr_002ffactorial_005fexpand"></a><dl>
<dt><a name="index-factorial_005fexpand"></a>Option variable: <strong>factorial_expand</strong></dt>
<dd><p>Default value: false
</p>
<p>The option variable <code>factorial_expand</code> controls the simplification of
expressions like <code>(x+n)!</code>, where <code>n</code> is an integer.
See <code><a href="#factorial">factorial</a></code> for an example.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Gamma-and-factorial-functions">Gamma and factorial functions</a>
·</div></dd></dl>
<a name="genfact"></a><a name="Item_003a-MathFunctions_002fdeffn_002fgenfact"></a><dl>
<dt><a name="index-genfact"></a>Function: <strong>genfact</strong> <em>(<var>x</var>, <var>y</var>, <var>z</var>)</em></dt>
<dd>
<p>Returns the generalized factorial, defined as
<code>x (x-z) (x - 2 z) ... (x - (y - 1) z)</code>. Thus, when <var>x</var> is an integer,
<code>genfact (x, x, 1) = x!</code> and <code>genfact (x, x/2, 2) = x!!</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Gamma-and-factorial-functions">Gamma and factorial functions</a>
·</div></dd></dl>
<a name="minfactorial"></a><a name="Item_003a-MathFunctions_002fdeffn_002fminfactorial"></a><dl>
<dt><a name="index-minfactorial"></a>Function: <strong>minfactorial</strong> <em>(<var>expr</var>)</em></dt>
<dd>
<p>Examines <var>expr</var> for occurrences of two factorials
which differ by an integer.
<code>minfactorial</code> then turns one into a polynomial times the other.
</p>
<div class="example">
<pre class="example">(%i1) n!/(n+2)!;
n!
(%o1) --------
(n + 2)!
(%i2) minfactorial (%);
1
(%o2) ---------------
(n + 1) (n + 2)
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Number-theory">Number theory</a>
·</div></dd></dl>
<a name="sumsplitfact"></a><a name="Item_003a-MathFunctions_002fdefvr_002fsumsplitfact"></a><dl>
<dt><a name="index-sumsplitfact"></a>Option variable: <strong>sumsplitfact</strong></dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>sumsplitfact</code> is <code>false</code>,
<code><a href="#minfactorial">minfactorial</a></code> is applied after a <code><a href="#factcomb">factcomb</a></code>.
</p>
<div class="example">
<pre class="example">(%i1) sumsplitfact;
(%o1) true
</pre><pre class="example">(%i2) n!/(n+2)!;
n!
(%o2) --------
(n + 2)!
</pre><pre class="example">(%i3) factcomb(%);
n!
(%o3) --------
(n + 2)!
</pre><pre class="example">(%i4) sumsplitfact: not sumsplitfact ;
(%o4) false
</pre><pre class="example">(%i5) n!/(n+2)!;
n!
(%o5) --------
(n + 2)!
</pre><pre class="example">(%i6) factcomb(%);
1
(%o6) ---------------
(n + 1) (n + 2)
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Gamma-and-factorial-functions">Gamma and factorial functions</a>
·<a href="maxima_424.html#Category_003a-Simplification-flags-and-variables">Simplification flags and variables</a>
·</div></dd></dl>
<a name="Item_003a-MathFunctions_002fnode_002fRoot-Exponential-and-Logarithmic-Functions"></a><hr>
<div class="header">
<p>
Next: <a href="maxima_51.html#Root-Exponential-and-Logarithmic-Functions" accesskey="n" rel="next">Root Exponential and Logarithmic Functions</a>, Previous: <a href="maxima_49.html#Functions-for-Complex-Numbers" accesskey="p" rel="previous">Functions for Complex Numbers</a>, Up: <a href="maxima_47.html#Elementary-Functions" accesskey="u" rel="up">Elementary Functions</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
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