= Open-string oscillator degeneracy
{title2=$F(w)=\prod_{n\ge1}(1-w^n)^{-24}$}
For an ordinary critical <open string> with all <Neumann boundary conditions>, there are 24 independent transverse creation oscillators at each positive mode number $n$. Summing their nonnegative occupation numbers gives one factor $(1-w^n)^{-24}$. The coefficient $d_p=[w^p]F(w)$ counts oscillator polarizations at level $p$, excluding continuous center-of-mass momentum and additional endpoint multiplicities. <Cauchy coefficient formula> gives $d_p=(2\pi i)^{-1}\oint F(w)w^{-p-1}dw$.
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