The nice question by Mr. Oloa A closed form for the series $\sum_{n=1}^{\infty} \frac{H_n^2-(\gamma + \ln n)^2}{n}$ made me think of another question that is similar to that one , namely
$$\sum_{n=1}^{\infty} \frac{(H_n-\gamma)^2-\log^2(n)}{n}$$
As in the presented proof there I might think o...
I have found a closed form for the following new series involving non-linear harmonic numbers.
Proposition. $$\sum_{n=1}^{\infty} \dfrac{H_n^2-(\gamma + \ln n)^2}{n} = \dfrac{5}{3}\zeta(3)-\dfrac{2}{3}\gamma^3-2\gamma \gamma_{1}-\gamma_{2} $$ where \begin{align} & H_{n}: =\sum_{k=1}...
In the context of combinatorial mathematics, stars and bars is a graphical aid for deriving certain combinatorial theorems. It was popularized by William Feller in his classic book on probability. It can be used to solve many simple counting problems, such as how many ways there are to put n indistinguishable balls into k distinguishable bins.
== Statements of theorems ==
=== Theorem one ===
For any pair of positive integers n and k, the number of distinct k-tuples of positive integers whose sum is n is given by the binomial coefficient .
=== Theorem two ===
For any pair of natural nu...
