Search Results

Let $H_k=\sum_{j=1}^k\frac1j$ be the harmonic numbers. QUESTION. Can you find an evaluation of the following sum? $$\sum_{a=1}^b(-1)^a\binom{n}{b-a}\frac{H_{b-a}}a.$$

Fix an integer $k\geq3$. Define the two families of sequences $\{x_n\}$ and $\{y_n\}$ according to the rules: $$x_n=\frac{x_{n-1}^2+x_{n-2}^2+\cdots+x_{n-k+1}^2}{x_{n-k}} \qquad n\geq k$$ and $$y_n=\f …

Consider the following sequence defined as a sum $$a_n=\sum_{k=0}^{n-1}\frac{3^{3n-3k-1}\,(7k+8)\,(3k+1)!}{2^{2n-2k}\,k!\,(2k+3)!}.$$ QUESTION. For $n\geq1$, is the sequence of rational numbers $a_n$ …

The sequence $a_n=\sum_{k=0}^n\binom{n}k^24^k$ is listed on OEIS along with a couple of combinatorial interpretations. What interested me at the moment is the plethora of binomial single-sums for the …

We say two permutations $\pi_1$ and $\pi_2$ in the symmetric group $\mathfrak{S}_n$ are $k$-equivalent, denoted $\pi_1 \sim_k \pi_2$, if one can be determined from the other after a finite number of …

The integer partition function $p(n)$ has a generating function given by $$\frac1{(q)_{\infty}}=\sum_{n=0}^{\infty}p(n)q^n$$ with $(q)_{\infty}=\prod_{m=1}^{\infty}(1-q^m)$. The long-standing problem …

Let $x$ be a variable. Define the following family of sequences (reminiscent of Lucas polynomials) according to the rule: $P_0(x):=0, P_1(x):=1$ and for $n\geq2$ by $$P_n(x)=xP_{n-1}(x)-P_{n-2}(x).$$ …