Summation with binomial coefficient
by fungarwai, Jan 12, 2019, 2:04 PM
Summation with one binomial coefficient
from Binomial theorem
Proof
Proof
Proof
Proof
Put
Proof
Proof
Proof
Proof
which is Hockey-stick identity
Other forms
Summation with two binomial coefficient
Proof with Generating function
which is Vandermonde's identity
Proof with Generating function
Proof with Hypergeometric function
Example
Summation with three binomial coefficient
which is the Li Shanlan identity
Proof with Hypergeometric function
Summation with multiple binomial coefficient
where
It is the Generalized Vandermonde's identity
Proof
Named as Vandermonde type convolution formula in
Combinatorial identities associated with new families of the numbers and polynomials and their approximation values
(Theorem 5.4 of Section 5)
The probability of the sum of negative binomial random variables also implies that
Reference: Negative binomial distribution - sum of two random variables
Proof by recalculating number of integer solution of a linear equation
Proof by generating functions of negative binomial series
Proof by sum of negative binomial random variables
It can be further generalised as:
Proof by generating functions of negative binomial series
Example
from Binomial theorem
Proof
Proof
Proof
Proof
Put
Proof
Proof
Proof
Proof
which is Hockey-stick identity
Other forms
Summation with two binomial coefficient
Proof with Generating function
which is Vandermonde's identity
Proof with Generating function
Proof with Hypergeometric function
Example
Summation with three binomial coefficient
which is the Li Shanlan identity
Proof with Hypergeometric function
Summation with multiple binomial coefficient
where
It is the Generalized Vandermonde's identity
Proof
expand to get the result
Named as Vandermonde type convolution formula in
Combinatorial identities associated with new families of the numbers and polynomials and their approximation values
(Theorem 5.4 of Section 5)
The probability of the sum of negative binomial random variables also implies that
Reference: Negative binomial distribution - sum of two random variables
Proof by recalculating number of integer solution of a linear equation
for
There are solutions for
On one hand,
On the other hand,
There are solutions for
On one hand,
On the other hand,
Proof by generating functions of negative binomial series
Proof by sum of negative binomial random variables
Let , the moment generating function of is
It can be further generalised as:
Proof by generating functions of negative binomial series
Example
This is a question sent from Taiharward on Nov 7, 2024 and posted at
https://artofproblemsolving.com/community/c4h3440954_evaluate_summation
Q: Find , where are triples of positive integers.
A:
https://artofproblemsolving.com/community/c4h3440954_evaluate_summation
Q: Find , where are triples of positive integers.
A:
for n in range(0,30+1): s=0 for x in range(0,n+1): for y in range(0,int(n/2)+1): for z in range(0,int(n/3)+1): if x+2*y+3*z==n: s=s+x*y*z print("n=",n,"sum=",s)
This post has been edited 24 times. Last edited by fungarwai, Yesterday at 12:54 AM
by cardesigner06, Apr 30, 2019, 6:41 PM