PascalBinomial

PascalBinomial[n,m]

gives the binomial coefficient that preserves Pascal's identity.

Details

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

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Basic Examples (5)

Evaluate numerically:

Evaluate symbolically:

Construct Pascal's triangle:

Plot over a subset of the reals as a function of its first parameter:

Plot over a subset of the reals as a function of its second parameter:

Plot over a subset of the complexes:

Scope (36)

Numerical Evaluation (7)

Evaluate numerically:

Evaluate for half-integer arguments:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number inputs:

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:

Values at zero:

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 :

Plot the real part of :

Plot the imaginary part of :

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:

Complex domain:

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:

is not injective:

is surjective:

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)

First derivative with respect to :

Higher derivatives with respect to :

Plot the higher derivatives with respect to for :

First derivative with respect to :

Series Expansions (4)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Find the series expansion at Infinity:

Find series expansion for an arbitrary symbolic direction :

Taylor expansion at a generic point:

Function Identities and Simplifications (2)

Functional identity:

Recurrence relations:

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:

Fractional binomial theorem:

Binomial coefficients mod 2:

Plot PascalBinomial in the arguments' plane:

Plot the logarithm of the number of ways to pick elements out of :

Compute higher derivatives of a product of two functions:

Properties & Relations (8)

On the integers, PascalBinomial[n,m] equals :

This can be expressed as $(-1)^m TemplateBox[{{-, n}, m}, Pochhammer]/m!$ 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:

Binomial coefficient mod :

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:

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PascalBinomial

Details

Examples

Basic Examples (5)