The binomial-coefficients tag has no wiki summary.

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### sum calculation

I would like to calculate, or bound from above, the following sum
$$
\sum_{i=0}^n(n-2i)^p{p \choose i},
$$
here $p\geq 2$.
Any references are very welcome.
Thank you.

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votes

**1**answer

68 views

### Bounded convolutions with binomial coefficients

I need to figure out a nice family of decaying functions such that
$\sum_{d=2}^k {k \choose d} f_k(d) \leq 1/k$ and $f_k(d)\geq f_k(d+1)$
How can I figure out what good candidates could be?
Any ...

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295 views

### Polynomials with presumably positive coefficients

The $q$-Pochhammer symbol
$(q) _ 0:=1$ and $(q) _ n:=\prod _ {j=1}^n (1-q^j)$ for $n > 0$ is clearly a polynomial in $q$ which has both positive and negative coefficients when $n>0$.
The ...

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445 views

### Does anyone know this sequence of polynomials?

A referee on a paper of mine showed me the following recurrence for polynomials $P_{n,k}\in\mathbb Q[q,q^{-1}]$ for $n\geq 0$ and $0\leq k\leq n/2$.
...

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### Inverse of a matrix with binomial coefficients

Let $a(n,k)=(-1)^k {{2n-k}\choose k}$ for $0 \le k \le n$ and $a(n,k)=0$ else. Then it is known (cf. OEIS A005439 and A098435) that the first column of the inverse matrix of $(a(i,j))_{i,j\ge0}$ is ...

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196 views

### Double sum involving binomial coefficients

I came across a sum of binomial coefficients while trying to solve a problem involving $SU(2)$ group integrals. I am not able to solve it, nor I found a similar identity in the literature. I would ...

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209 views

### Recurrence relation for trinomial Apery numbers

It is well known (Beukers 1987) that the Apery numbers $$A_n\equiv A_n^{(2)}=\sum\limits_{k=0}^n\binom{n}{k}^2\binom{n+k}{k}^2$$ satisfy the fancy recurrence relation
...

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66 views

### alternating sum with Barnes G functions

Let $B(n)=(n-2)!(n-3)!\cdots 1!$ denote the Barnes G-function.
I am pretty sure that
$$
\sum_{m=0}^{k^2-1}
(-1)^m\binom{k^2-1}m
\frac{G(k+n-m+1)}{G(n-m+1)G(k+1)(k^2)!}
= n-2k^2-2k
$$ when $k$ is ...

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232 views

### A coincidence concerning Fermat primes, binomial sums, and eta quotients?

Let $w_k$ be a primitive k th root of unity, where k is a power of 2. In response to a question, Robert Israel gave the solution,
$$\sum_{n=0}^\infty \frac{(-1)^n}{\binom{kn}{kn/2}} = ...

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133 views

### Sum of binomial coefficients weighted by a lower incomplete regularized gamma function

The following summation turned up in the course of my research:
$$S_n=\sum_{k=0}^n {n \choose k}\lambda^k P(k,t)$$
where $P(k,t)=\frac{1}{\Gamma(k)}\int_{0}^t e^{-x}x^{k-1}dx$ is the lower ...

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164 views

### Weighted sum of ratio of partial sum of binomial coefficients

I would like to approximate the following sum when $n \rightarrow \infty$ and $n \gg k$,
$\sum_{x = k}^n \sum_{y > x}^n \frac{\sum_{m = 0}^{k - 1} {y - 2 \choose m}}{\sum_{m = 0}^k {y - 1 \choose ...

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151 views

### An extrasensory perception strategy :-)

I asked this question at MSE some months ago
but I received only partial answers, so I put it here. The following sounds nice for me and I spent a good time during the investigation. But I am a ...

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vote

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128 views

### Asymptotic behaviour of sequence

I am interested in the sequence
$$a(n)=\sum_{k=0}^n {p(n-k)-1 \choose k}$$
where $p(n)=(r-1)n^2+(2r-1)n+r$ for some $r \in \mathbb{N}$ or more generally any polynomial equation.
When $r=1$ this ...

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vote

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288 views

### Another identity involving sums of (alternating) binomial coefficients.

I have derived two different solutions to the same problem involving computing the expected time to find $k$ swaps when collecting coupons. However to me the two sums, although apparently numerically ...

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70 views

### binomial transform, Hurwitz zeta function

For $j,n\in\mathbb Z_+$,
let
$$
L_{j,n}^{(t)}=
\sum_{m=0}^{n} \Bigl(-\frac 12\Bigr)^{n-m}{n\choose m}{m+j+1\choose m+1} \left(
\frac {1}{t+\frac 12}\right)^{m+j+2}
$$
and
$$
L_{j,n} ...

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68 views

### Congruences for generalized Franel numbers

Let us define generalized Franel numbers $f^{(m)}_n$ through recurrence relations:
$f^{(1)}_n=1$ for all $n$, and $$f^{(m)}_n=\sum\limits_{k=0}^n\binom{n}{k}^3f^{(m-1)}_k.$$ In fact ...

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81 views

### Simplifying a finite difference sum that represents a model posterior to avoid numerical issues.

Problem
Is it possible to simplify/rewrite the following expression, preferably without explicit sums, such that it can be computed without numerical issues when the $n_*$ are in the range of ...

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373 views

### Calculating the Shapley value in a weighted voting game.

Given a special case of WVG (Weighted Voting Game) of $a$ 1s and $b$ 2s and a quota q, $ [q:1,1,1,1..1,2,2,..2] $. I need help with calculating the Shapley value of a player with a weight of $2$ and a ...

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171 views

### Asymtotic Complexity Analysis using logarithms and binomial coefficients

On page 11 of "Smaller decoding exponents: ball-collision decoding" by Berstein et.al. they have the formula \begin{equation}\lim_{n \rightarrow \infty} ...

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166 views

### $q$-binomials in terms of ordinary binomials

Is there a known identity that expresses the $q$-binomial coefficients (also known as Gaussian binomial coefficients) in terms of the (ordinary) binomial coefficients?