Infinite series involving harmonic numbers and reciprocal of binomial coefficients

Yamamoto's integral was the integral associated with 2-posets, which was first introduced by Yamamoto. In this paper, we obtained the values of infinite series involving harmonic numbers and reciprocal of binomial coefficients by using some techniques of Yamamoto's integral. We determine the value of infinite series of the form:

$$\sum _{m_1,\dots ,m_n,{\ell }_1,\dots ,{\ell }_k\ge 1}\frac{H_{m_1}^{(a_1)}\cdots H_{m_n}^{(a_n)}}{m_1^{b_1}\cdots m_n^{b_n}{\ell }_1^{c_1}\cdots {\ell }_k^{c_k}(\genfrac{}{}{0ex}{}{m_1+\cdots +m_n+{\ell }_1+\cdots +{\ell }_k}{{\ell }_k})},$$

in terms of a finite sum of multiple zeta values, for positive integers $a_1,\dots ,a_n,b_1,\dots ,b_n,c_1,\dots ,c_k$.