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Template:Binomial/doc
From OeisWiki
binomial}}, {{binom}} or {{C(n,k)}} mathematical formatting template and/or mathematical function template returns either the typeset (HTML+CSS or LaTeX) expression of the the binomial coefficient or the numerical result, for nonnegative integers
and
, of
.
where
is the falling factorial.
(rounding error).
The {{n |
k |
( nk ) |
Contents
Usage
- {{binomial|nonnegative integer n|nonnegative integer k|format}}
or
- {{binom|nonnegative integer n|nonnegative integer k|format}}
or
- {{C(n,k)|nonnegative integer n|nonnegative integer k|format}}
where format is
- htm: HTML+CSS text style (default), or
- HTM: HTML+CSS display style, or
- tex: LaTeX text style, or
- TEX: LaTeX display style, or
- #: numerical result.
Examples
The code
: before {{math|{{binom|''n''|''k''|htm}}|&}} after
yields the text style HTML+CSS
- before
after( nk )
The code
: before {{math|{{binom|''n''|''k''|tex}}|$}} after
yields the text style LaTeX
- before after
The code
{{indent}}{{math|{{binom|''n''|''k''|HTM}}|&&}}
yields the display style HTML+CSS
(
|
The code
: {{math|{{binom|''n''|''k''|TEX}}|$$}}
yields the display style LaTeX
The code
: before {{math|{{binom|9|3|htm}} {{=}} {{binom|9|3|#}}|&}} after
yields the text style HTML+CSS
- before
after( 93 ) = 84
The code
: before {{math|{{binom|9|3|tex}} {{=}} {{binom|9|3|#}}|$}} after
yields the text style LaTeX
- before after
Examples with numerical result
The algorithm for the evaluation of binomial coefficients uses
(
|
where
n (m) :=
(n − i ) |
Examples with numerical result: valid input
Numerical tests.— Daniel Forgues 21:55, 11 March 2018 (EDT) Code Result Comment Code Result Comment {{binomial|0|0|#}} 1 {{binom|0|1|#}} 0 {{binomial|1|1|#}} 1 {{binom|2|0|#}} 1 {{binomial|2|1|#}} 2 {{binom|2|2|#}} 1 {{C(n,k)|3|1|#}} 3 {{C(n,k)|4|2|#}} 6 {{C(n,k)|5|3|#}} 10 {{C(n,k)|6|2|#}} 15 {{C(n,k)|7|3|#}} 35 {{C(n,k)|8|3|#}} 56 {{C(n,k)|9|3|#}} 84 {{C(n,k)|10|3|#}} 120 {{C(n,k)|11|4|#}} 330 {{C(n,k)|12|4|#}} 495 {{C(n,k)|13|5|#}} 1287 {{C(n,k)|14|5|#}} 2002 {{C(n,k)|15|3|#}} 455 {{C(n,k)|16|3|#}} 560 {{C(n,k)|17|3|#}} 680 {{C(n,k)|17|23|#}} 0 {{C(n,k)|30|21|#}} 14307150 {{C(n,k)|47|13|#}} 140676848445 {{C(n,k)|48|19|#}} 11541847896480 {{C(n,k)|49|19|#}} 18851684897585 ( 4919 ) = 18851684897584 (rounding error)* {{C(n,k)|80|3|#}} 82160 {{C(n,k)|83|3|#}} 91881 {{C(n,k)|99|31|#}} 4.5764000431736E+25 {{C(n,k)|91|3|#}} 121485
( 4919 ) = 49(30) / 19! = 2293215102242267478449061888000 / 121645100408832000 = 18851684897584 |
Examples with numerical result: invalid input
Code Result {{C(n,k)|-1|3|#}} C(n,k) error: First and second arguments, n and k, must be nonnegative integers, with k up to 32. {{C(n,k)|0.5|3|#}} C(n,k) error: First and second arguments, n and k, must be nonnegative integers, with k up to 32. {{C(n,k)|text|3|#}} C(n,k) error: First and second arguments, n and k, must be nonnegative integers, with k up to 32. {{C(n,k)|6 blobs|3|#}} C(n,k) error: First and second arguments, n and k, must be nonnegative integers, with k up to 32. {{C(n,k)|12|nine|#}} C(n,k) error: First and second arguments, n and k, must be nonnegative integers, with k up to 32. {{C(n,k)|99|43|#}} C(n,k) error: First and second arguments, n and k, must be nonnegative integers, with k up to 32. {{C(n,k)|99|33|#}} C(n,k) error: First and second arguments, n and k, must be nonnegative integers, with k up to 32.