In  [Uniform Approach to Double Shuffle and Duality Relations of Various q-Analogs of Multiple Zeta Values via Rota-Baxter Algebras][1], Proposition 10.8, Jianqiang Zhao mentiones the sequence:
$$a_n=\sum_{1\leq d,k\leq n}\sum_{\substack{x_1+\dots+x_d=d+k-1\\1\leq x_1,\dots,x_d}}x_1 x_2\cdots x_d. $$
(Actually under the first sum it reads $1\leq d\leq k\leq n$, but this is probably a typo, since in Table 8 on the same page, the sequence starts with $1,8,49,\dots$, which is what you get when you sum over  ${1\leq d,k\leq n}$; see also the oeis-entry: https://oeis.org/A261668.)

Numerically, from the first few values it seems like the asymptotic of this sequence might be
$$a_n= \frac{\sqrt{3} \left(\frac{27}{4}\right)^{n}}{\sqrt{\pi} \sqrt{n}} \left(\frac{81}{40} -\frac{10107}{8000 \, n} + \frac{3660889}{3200000 \, n^{2}} - \frac{3852905951}{3456000000 \, n^{3}} + \frac{27568631693309}{24883200000000 \, n^{4}} - \frac{49495499873402443}{44789760000000000 \, n^{5}} + \frac{533089133599474131889}{483729408000000000000 \, n^{6}} - \frac{957527162102649890515621}{870712934400000000000000 \, n^{7}} + \frac{2757322129178663878655381987}{2507653251072000000000000000 \, n^{8}} - \frac{44721988445925710711255052650633}{40623982667366400000000000000000 \, n^{9}} \pm O\left(\frac{1}{n^{10}}\right)\right)$$

Has it been proved somewhere, that this is the correct asymptotic?


  [1]: http://arxiv.org/abs/1412.8044