I have a question. If I define the multifold convolution of Harmonic numbers as $\sum_{n_1=1}^{\infty} \cdots \sum_{n_k=1}^{\infty} H_{n_1} \cdots H_{n_k} \mathbf{1}_{\{n\}}(n_1+\dots+n_k)$ for the $k$-fold convolution, $n$th term, is there any known formula, even one that involves a small number of summands?
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1$\begingroup$ I mean there is the formula $\sum_{m_1,\dots ,m_k=1}^\infty \frac{1}{m_1\dots m_k} \binom{n+k-1-m_1 - \dots - m_k }{k-1}$ which has more terms but is dominated by the ones with $m_1,\dots, m_k$ relatively small. $\endgroup$– Will SawinCommented Oct 11 at 20:17
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$\begingroup$ Thanks. By the way, I should have said a small number of ``summations'' not summands. But your answer shows promise. (So you get this sort-of like "stars-and-bars"?) I could use it in the form $\sum_{r} N(n,k,r)/r$ where $N(n,k,r)$ is the sum over all $m_1,\dots,m_k$ such that $m_1\cdots m_k=r$ of the quantity $\binom{n+k-1-m_1-\dots-m_k}{k-1}$. Interesting. $\endgroup$– Shannon StarrCommented Oct 12 at 21:18
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