# Category:Binomial Coefficients

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This category contains results about **Binomial Coefficients**.

Definitions specific to this category can be found in Definitions/Binomial Coefficients.

Let $n \in \Z_{\ge 0}$ and $k \in \Z$.

Then the **binomial coefficient** $\dbinom n k$ is defined as:

- $\dbinom n k = \begin{cases} \dfrac {n!} {k! \paren {n - k}!} & : 0 \le k \le n \\ & \\ 0 & : \text { otherwise } \end{cases}$

where $n!$ denotes the factorial of $n$.

## Subcategories

This category has the following 46 subcategories, out of 46 total.

## Pages in category "Binomial Coefficients"

The following 137 pages are in this category, out of 137 total.

### A

- Alternating Sum and Difference of Binomial Coefficients for Given n
- Alternating Sum and Difference of Binomial Coefficients for Given n/Corollary
- Alternating Sum and Difference of r Choose k up to n
- Alternating Summation of Binomial Coefficient of Summation of Binomial Coefficient of Sequence
- Approximation to 2n Choose n
- Approximation to x+y Choose y

### B

- Babbage's Congruence
- Binomial Coefficient 2 n Choose n is Divisible by All Primes between n and 2 n
- Binomial Coefficient expressed using Beta Function
- Binomial Coefficient involving Power of Prime
- Binomial Coefficient involving Prime
- Binomial Coefficient is instance of Gaussian Binomial Coefficient
- Binomial Coefficient is Integer
- Binomial Coefficient n Choose j in terms of n-2 Choose r
- Binomial Coefficient n Choose k by n Plus 1 Minus n Choose k + 1
- Binomial Coefficient of Prime
- Binomial Coefficient of Prime Minus One Modulo Prime
- Binomial Coefficient of Prime Plus One Modulo Prime
- Binomial Theorem
- Binomial Theorem/Abel's Generalisation

### C

- Charles Babbage's Conjecture
- Chu-Vandermonde Identity
- Chu-Vandermonde Identity/Falling Factorial Variant
- Chu-Vandermonde Identity/Rising Factorial Variant
- Condition for Equality of Adjacent Binomial Coefficients
- Condition for Increasing Binomial Coefficients
- Condition on Lower Coefficient for Binomial Coefficient to be Maximum

### D

### F

- Factors of Binomial Coefficient
- Factors of Binomial Coefficient/Complex Numbers
- Factors of Binomial Coefficient/Corollary 1
- Factors of Binomial Coefficient/Corollary 2
- Fibonacci Number as Sum of Binomial Coefficients
- Fibonacci Number by Power of 2
- Fibonacci Number plus Binomial Coefficient in terms of Fibonacci Numbers
- Fibonacci Number with Prime Index 2n+1 is Congruent to 5^n Modulo p

### I

### L

### M

### N

- N Choose k is not greater than n^k
- N Choose Negative Number is Zero
- Negated Upper Index of Binomial Coefficient
- Negated Upper Index of Binomial Coefficient/Corollary 1
- Negated Upper Index of Binomial Coefficient/Corollary 2
- Numbers Expressed as Sums of Binomial Coefficients
- Numbers with Square-Free Binomial Coefficients
- Numbers with Square-Free Binomial Coefficients/Lemma

### P

### R

### S

- Sequence of Binomial Coefficients is Strictly Decreasing from Half Upper Index
- Sequence of Binomial Coefficients is Strictly Increasing to Half Upper Index
- Square Formed from Sum of 4 Consecutive Binomial Coefficients
- Sum of 4 Consecutive Binomial Coefficients forming Square
- Sum of Bernoulli Numbers by Binomial Coefficients Vanishes
- Sum of Binomial Coefficients over Lower Index
- Sum of Binomial Coefficients over Lower Index/Corollary
- Sum of Binomial Coefficients over Upper Index
- Sum of Euler Numbers by Binomial Coefficients Vanishes
- Sum of Even Index Binomial Coefficients
- Sum of k Choose m up to n
- Sum of Odd Index Binomial Coefficients
- Sum of r+k Choose k up to n
- Sum of Sequence of Binomial Coefficients by Powers of 2
- Sum of Sequence of n Choose 2
- Sum of Sequence of Product of Fibonacci Number with Binomial Coefficient
- Sum of Squares of Binomial Coefficients
- Sum over j, k of -1^j+k by j+j Choose k+l by r Choose j by n Choose k by s+n-j-k Choose m-j
- Sum over k from 1 to n of n Choose k by Sine of n Theta
- Sum over k of -1^k by n choose k by r-kt choose n by r over r-kt
- Sum over k of -2 Choose k
- Sum over k of m choose k by -1^m-k by k to the n
- Sum over k of m Choose k by k minus m over 2
- Sum over k of m+r+s Choose k by n+r-s Choose n-k by r+k Choose m+n
- Sum over k of m-n choose m+k by m+n choose n+k by Stirling Number of the Second Kind of m+k with k
- Sum over k of m-n choose m+k by m+n choose n+k by Unsigned Stirling Number of the First Kind of m+k with k
- Sum over k of n Choose k by Fibonacci Number with index m+k
- Sum over k of n Choose k by Fibonacci t to the k by Fibonacci t-1 to the n-k by Fibonacci m+k
- Sum over k of n Choose k by p^k by (1-p)^n-k by Absolute Value of k-np
- Sum over k of n Choose k by x to the k by kth Harmonic Number
- Sum over k of n Choose k by x to the k by kth Harmonic Number/x = -1
- Sum over k of n+k Choose 2 k by 2 k Choose k by -1^k over k+1
- Sum over k of n+k Choose m+2k by 2k Choose k by -1^k over k+1
- Sum over k of r Choose k by -1^r-k by Polynomial
- Sum over k of r Choose k by Minus r Choose m Minus 2k
- Sum over k of r Choose k by s Choose k by k
- Sum over k of r Choose k by s+k Choose n by -1^r-k
- Sum over k of r Choose k by s-kt Choose r by -1^k
- Sum over k of r Choose m+k by s Choose n+k
- Sum over k of r Choose m+k by s Choose n+k/Proof 1
- Sum over k of r+tk choose k by s-tk choose n-k
- Sum over k of r-k Choose m by s Choose k-t by -1^k-t
- Sum over k of r-k Choose m by s Choose k-t by -1^k-t/Proof 1
- Sum over k of r-k Choose m by s+k Choose n
- Sum over k of r-k Choose m by s+k Choose n/Proof 1
- Sum over k of r-kt Choose k by r over r-kt by s-(n-k)t Choose n-k by s over s-(n-k)t
- Sum over k of r-kt choose k by r over r-kt by z^k
- Sum over k of r-kt choose k by z^k
- Sum over k of r-tk Choose k by s-t(n-k) Choose n-k by r over r-tk
- Sum over k of Stirling Number of the Second Kind of n+1 with k+1 by Unsigned Stirling Number of the First Kind of k with m by -1^k-m
- Sum over k of Stirling Numbers of the Second Kind of k with m by n choose k
- Sum over k of Stirling Numbers of the Second Kind of k+1 with m+1 by n choose k by -1^k-m
- Sum over k of Unsigned Stirling Numbers of the First Kind of n with k by k choose m
- Sum over k of Unsigned Stirling Numbers of the First Kind of n+1 with k+1 by k choose m by -1^k-m
- Sum over k to n of k Choose m by kth Harmonic Number
- Summation Formula for Reciprocal of Binomial Coefficient
- Summation from k to m of 2k-1 Choose k by 2n-2k Choose n-k by -1 over 2k-1
- Summation from k to m of r Choose k by s Choose n-k by nr-(r+s)k
- Summation to n of Power of k over k
- Summations of Products of Binomial Coefficients
- Symmetry Rule for Binomial Coefficients