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An extended comment that has a chance of being useful. First, note that if you interchange $a_k$ and $a_{k+1}$ then only the $k$-th term on the left side changes and the right side doesn't change at …

This is not an answer, but I want to use displays. What you are asking is whether the coefficients of the taylor expansion of $$\left( \sum_{k=0}^\infty \frac{x^k}{(N-1)k+1} \right)^N$$ are all at mo …

On the basis of rather convincing numerical evidence (iterative optimisation that always converges to the same place regardless of starting point), I conjecture that the worst case is $$ a_i = \frac{i …

Iosif's answer is very interesting and if anyone knows how to do the same thing in Maple I'd like to know. However I disagree with Iosif about the difficulty. Mathematica will use a systematic proced …

If you want a more precise value you can expand $k^{-s}$ around $k=N/2$ and sum term by term. I get $$\frac{2^N}{(N/2)^s}\Bigl(1 + \frac{s(1+s)}{2N} + \frac{s(1+s)(2+s)(3+s)}{8N^2} + O(N^{-3})\Bigr). …

Since the question is about compositions, there is a generating function approach when $\phi(a_1,\ldots,a_n)=\prod_i f(a_1)$ for some function $f$. Namely, $S(n,k)$ is the coefficient of $x^n$ in $\l …

Sketch: Write $p=1/x$, so that you want to show $f(x) = \sin(\pi/4+\pi x)-1/2-\pi x^2\ge 0$ for $0\le x\le 0.355$ approximately. First, $f''(x)<0$ in $[0,1/2]$ (by a large margin), so $f'(x)$ is stric …

This paper is relevant: E. Mailhot, Une propriété de la variance de certaines lois de probabilité réelles tronqées, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985) 241–244. Mailhot proves that the va …

Since the case $x=1$ is trivial, assume $x>1$ and divide both sides by $x-1$. It becomes $$(x+1)^{p-1}\ge \frac{x^p-1}{x-1} = 1 + x + x^2 + \cdots + x^{p-1}.$$ Expand the left side by the binomial the …

Let $\boldsymbol{x}=(x_1,\ldots,x_n)$ be a real vector. Define the normalized power-sum symmetric polynomials by $\pi_j(\boldsymbol{x})=\frac 1n(x_1^j+\cdots+x_n^j)$. For a partition $\lambda= (j_1,\l …

If the value of the sum is $\frac13-\varDelta(k)$, then it appears that $\varDelta(k)$ satisfies the recurrence $$ (8k+4)\varDelta(k) = (7k-5)\varDelta(k-1) + k\varDelta(k-2).$$ Note that I didn't pro …

Taking $f(a,b)=\frac{a+b}{a}$ as in the other question, I conjecture that the optimal $C$ is $e$. One bit of evidence is that the limit of the ratio for $a=cb$ and $b\to\infty$ is $(c+1)^{1/c}$ which …