# Tagged Questions

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### Alternating sum of square roots of binomial coefficients

Let $$c_n = \sum_{r=0}^n (-1)^r \sqrt{\binom{n}{r}}.$$ It is clear that $c_n = 0$ if $n$ is odd. Remarkably, it appears that despite the huge positive and negative contributions in the sum ...
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### Is the sum $\sum\limits_{j=0}^{k-1}(-1)^{j+1}(k-j)^{2k-2} \binom{2k+1}{j} \ge 0?$

I am trying to prove $\sum\limits_{j=0}^{k-1}(-1)^{j+1}(k-j)^{2k-2} \binom{2k+1}{j} \ge 0$. This inequality has been verified by computer for $k\le40$. Some clues that might work (kindly provided by ...
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### Proving $\sum_{k=0}^{2m}(-1)^k{\binom{2m}{k}}^3=(-1)^m\binom{2m}{m}\binom{3m}{m}$

I found the following formula in a book without any proof: $$\sum_{k=0}^{2m}(-1)^k{\binom{2m}{k}}^3=(-1)^m\binom{2m}{m}\binom{3m}{m}.$$ This does not seem to follow immediately from the basic ...
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### A binomial generalization of the FLT: Bombieri's Napkin Problem

This is an extract from Apéry's biography (which some of the people have already enjoyed in this answer). During a mathematician's dinner in Kingston, Canada, in 1979, the conversation turned ...
I have veriﬁed the following double sum is always an integer for $s$ up to $1000$ via Maple. But I can not prove it. Proofs, hints, or references are all welcome. Thanks! $$\sum_{m=s}^{2s}\sum_{k=0}^{... 3answers 520 views ### Is there a closed formula for the generating function of some trinomial coefficients? We learn in calculus how to obtain a sum of binomial coefficients \frac{(2d)!}{(d!)^2} in terms of a generating function \sum_{d \geq 0} \frac{(2d)!}{(d!)^2} x^d by the Taylor series of (1-4x)^... 0answers 290 views ### Sum of binomial coefficients weighted by a lower incomplete regularized gamma function The following summation turned up in the course of my research:$$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 ...
..I wonder if the following formula can be calculated? $\sum_{k=0}^m {m \choose k} {2k \choose n}$