Sum of $\sum\limits_{m = 1}^\infty {\frac{{m(m + 1)}}{2}} {p^{\frac{{m(m - 1)}}{2}}}\left( {1 - {p^m}} \right)$ I'm trying to find if the following sum is converging to some known identity:
$$\sum\limits_{m = 1}^\infty  {\frac{{m(m + 1)}}{2}} {p^{\frac{{m(m - 1)}}{2}}}\left( {1 - {p^m}} \right)$$
$$ p \in [{0,1}] $$
Thanks for any help

I am suggested to think that the answer is negative, because series of the form $\sum_{m=0}^\infty a_m p^{m^2}$ are not natural in the analysis, in contrast to series of the form $\sum_{m=0}^\infty a_m p^m$.
