PascalBinomial
PascalBinomial[n,m]
gives the binomial coefficient that preserves Pascal's identity.
Details
![](Files/PascalBinomial.en/details_1.png)
- Integer mathematical function, suitable for both symbolic and numerical manipulation.
- PascalBinomial is also known as combinations and as choose function.
- PascalBinomial gives the asymmetric coefficients that preserve Pascal's identity for all integer values. Use Binomial for coefficients that are symmetric for negative integer
. PascalBinomial and Binomial agree except for negative integer
.
- In general,
is defined by
or suitable limits of this.
- When
is a negative integer,
. »
- The particular limit chosen preserves Pascal's identity
for all complex
and
. »
- The symmetry rule
is satisfied for all
and most
but violated for negative integer
. »
- For integer arguments, PascalBinomial automatically evaluates to exact values.
- PascalBinomial is automatically evaluated symbolically for simple cases; FunctionExpand gives results for other cases. »
- PascalBinomial can be evaluated to arbitrary numerical precision.
- PascalBinomial automatically threads over lists.
- PascalBinomial can be used with Interval and CenteredInterval objects. »
Examples
open allclose allBasic Examples (5)
Scope (36)
Numerical Evaluation (7)
Evaluate for half-integer arguments:
The precision of the output tracks the precision of the input:
Evaluate efficiently at high precision:
Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:
Or compute average-case statistical intervals using Around:
Compute the elementwise values of an array:
Or compute the matrix PascalBinomial function using MatrixFunction:
Specific Values (5)
Values of PascalBinomial at particular points:
PascalBinomial for symbolic n:
Note that this is zero on all integers away from :
PascalBinomial[n,m] is zero whenever n and m are both negative integers:
Find a value of n for which PascalBinomial[n,2]=15:
Visualization (3)
Plot the PascalBinomial as a function of its parameter n:
Plot the PascalBinomial as a function of its parameter :
Function Properties (12)
Real domain of PascalBinomial as a function of its parameter n:
Real domain of PascalBinomial as a function of its parameter m:
Function range of PascalBinomial:
PascalBinomial has the mirror property :
Compute sums involving PascalBinomial:
When is positive,
is an analytic function of both variables:
This is not true for negative :
is neither non-decreasing nor non-increasing:
PascalBinomial is neither non-negative nor non-positive:
has singularities and discontinuities where
is a negative integer:
is neither convex nor concave:
TraditionalForm formatting:
Differentiation (3)
Series Expansions (4)
Applications (9)
There are ways to choose
elements without replacements from a set of
elements:
Check with direct enumeration:
There are ways to choose
elements with replacement from a set of
elements:
Check with direct enumeration:
There are ways to arrange
indistinguishable objects of one kind and
indistinguishable objects of another kind:
Illustrate the binomial theorem:
Plot PascalBinomial in the arguments' plane:
Plot the logarithm of the number of ways to pick elements out of
:
Properties & Relations (8)
On the integers, PascalBinomial[n,m] equals :
This can be expressed as for
and
otherwise:
An alternative formula on the integers:
Pascal's identity is satisfied everywhere:
In particular, it is true at the origin:
The symmetry rule may not hold for negative integer
:
It may hold for some values but will generally be violated for positive integer :
Binomial satisfies the symmetry rule everywhere:
PascalBinomial performs simple evaluations for symbolic arguments:
PascalBinomial typically does not evaluate when both arguments are symbolic:
Use FunctionExpand with conditions to achieve appropriate simplifications:
PascalBinomial[n,m] is 0 whenever n and m are both negative integers:
Use FullSimplify to simplify expressions involving binomial coefficients:
Use FunctionExpand to expand into Gamma functions:
Sums involving PascalBinomial:
Neat Examples (7)
Construct a graphical version of Pascal's triangle:
Extend the triangle to negative integers; unlabeled points indicate a zero value:
Binomial, by contrast, defines the top-left sector by reflecting the top-right sector:
Closed‐form inverse of Hilbert matrices:
Nested binomials over the complex plane:
Plot PascalBinomial at infinity:
Plot PascalBinomial for complex arguments:
Plot PascalBinomial at Gaussian integers:
Text
Wolfram Research (2024), PascalBinomial, Wolfram Language function, https://reference.wolfram.com/language/ref/PascalBinomial.html.
CMS
Wolfram Language. 2024. "PascalBinomial." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PascalBinomial.html.
APA
Wolfram Language. (2024). PascalBinomial. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PascalBinomial.html