The combinatorics of the leading root of the partial theta function

Recently Alan Sokal studied the leading root $x_0(q)$ of the partial theta function $Θ_0(x,q)=\sum\limits_{n=0}^\infty x^nq^{\binom n2}$, considered as a formal power series. He proved that all the coefficients of $-x_0(q)=1+q+2q^2+4q^3+9q^4+...$ are positive integers. I give here an explicit combinatorial interpretation of these coefficients. More precisely, I show that $-x_0(q)$ enumerates rooted trees that are enriched by certain polyominoes, weighted according to their total area.

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