  We derive an effective recursion for Witten's r-spin intersection numbers,
using Witten's conjecture relating r-spin numbers to the Gel'fand-Dikii
hierarchy (Theorem 4.1). Consequences include closed-form descriptions of the
intersection numbers (for example, in terms of gamma functions: Propositions
5.2 and 5.4, Corollary 5.5). We use these closed-form descriptions to prove
Harer-Zagier's formula for the Euler characteristic of M_{g,1}. Finally in
Section 6, we extend Witten's series expansion formula for the Landau-Ginzburg
potential to study r-spin numbers in the small phase space in genus zero. Our
key tool is the calculus of formal pseudodifferential operators, and is
partially motivated by work of Brezin and Hikami.
