# CoCoA:BinExp

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## BinExp

binomial expansion

### Description

This function computes the K-binomial expansion of N, i.e., the

unique expression

<verbatim>

``` N = Bin(N(K),K) + Bin(N(K-1),K-1) + ... + Bin(N(I),I)
```

</verbatim> where <formula>N(K) > ... > N(I) >= 1</formula>, for some I. The value returned is tagged for pretty printing.

It can also compute the sum of the binomial coefficients

appearing in the K-binomial expansion of N after replacing each

summand Bin(N(J),J) by Bin(N(J)+Up,J+Down). It is useful in generalizations of Macaulay's theorem characterizing Hilbert functions.

#### Example

```  BE := BinExp(13,4);
BE;
Bin(5,4) + Bin(4,3) + Bin(3,2) + Bin(1,1)
-------------------------------
BinExp(13,4,1,1);
16
-------------------------------
```

### Syntax

```BinExp(N:INT,K:INT):TAGGED(<quotes>\$binrepr.BinExp</quotes>)
BinExp(N:INT,K:INT,Up:INT,Down:INT):INT

where N and K are positive integers, and Up and Down are integers.
```

```   <type>integer</type>
```