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Here we investigate the $q$-series \begin{align*}\mathcal{U}_a(q)&=\sum_{n=0}^{\infty} MO(a;n)q^n&:=\sum_{0< k_1<k_2<\cdots<k_a}\frac{q^{k_1+k_2+\cdots+k_a}}{(1-q^{k_1})^2(1-q^{k_2})^2\cdots(1-q^{k_a})^2},\\\mathcal{U}_a^{\star}(q)&=\sum_{n=0}^{\infty}M(a;n)q^n&:=\sum_{1\leq k_1\leqk_2\leq\cdots\leq k_a}\frac{q^{k_1+k_2+\cdots+k_a}}{(1-q^{k_1})^2(1-q^{k_2})^2\cdots(1-q^{k_a})^2}.\end{align*} MacMahon introduced the $\mathcal{U}_a(q)$ in his seminal work onpartitions and divisor functions. Recent works show that these series are sumsof quasimodular forms with weights $\leq 2a.$ We make this explicit bydescribing them in terms of Eisenstein series. We use these formulas to obtainexplicit and general congruences for the coefficients $MO(a;n)$ and $M(a;n).$Notably, we prove the conjecture of Amdeberhan-Andrews-Tauraso as the $m=0$special case of the infinite family of congruences $$ MO(11m+10; 11n+7)\equiv0\pmod{11}, $$ and we prove that $$ MO(17m+16; 17n+15)\equiv 0\pmod{17}. $$ Weobtain further formulae using the limiting behavior of these series. For $n\leqa+\binom{a+1}2,$ we obtain a ``hook length'' formulae for $MO(a;n)$, and for$n\leq 2a$, we find that $M(a;n)=\binom{a+n-1}{n-a}+\binom{a+n-2}{n-a-1}.$

MacMahon's sums-of-divisors and allied $q$-series