All Questions
6
questions
2
votes
0
answers
77
views
Closed expression for a combinatorial sum
The following equality is true for every positive integer $n$ :
$$\sum_{k=0}^n {n \choose k} = 2^n $$
It is a special case ($p = 0$) of the sequence :
$$S_{p, n}=\sum_{k=0}^n k^p {n \choose k} $$
For ...
- 389
2
votes
0
answers
56
views
$f_n(ab) = f_n(a)f_n(b) + f_n(a-1)f_n(b-1) + f_n(a-2)f_n(b-2)+ ...+f_n(a-n)f_n(b-n)$
I was toying around when I noticed for $a,b > 0$:
$$f(ab) = f(a)f(b) + f(a-1)f(b-1)$$
is satisfied by $f(n) = T_n$ ; the triangular numbers $n(n+1)/2$.
This equation is not an addition formula, not ...
- 16.4k
2
votes
1
answer
241
views
Monomials in terms of binomial coefficients
Is there an explicit expression (or at least a recurrence relation) for the coefficients of a monomial $x^n$ in the basis of polynomials given by binomial coefficients $P_k(x) = \binom{x}{k}$, namely
$...
- 63
0
votes
1
answer
30
views
Is there a generalized formula for recusive Binomial coefficient?
Is there any way I can solve the following recursion function?
$$f(n) = \binom{f(n-1)}{2}$$
Or can be written as $$f(n) = 1/2(f(n-1)(f(n-1) - 1))$$
f(0) = 4, f(1) = 6, f(2) = 15, ...
- 367
2
votes
2
answers
119
views
An Identity for a Fibbonacci-Type Polynomial
Problem:
The polynomials $p_{n}\left(x\right)$ are defined recursively by the linear homogenous order 2 difference equation $$p_{n+1}\left(x\right)=2\left(1-2x\right)p_{n}\left(x\right)-p_{n-1}\left(...
- 124
3
votes
2
answers
625
views
Recurrence equation for central trinomial coefficients
I've come across the following exercise:
Give a recurrence equation for the central coefficients $(a_n)$, where for all $n$, $a_n$ is the coefficient of $X^n$ in $(1+X+X^2)^n$.
Here's what I've ...
- 759