Three proofs of the Goulden-Litsyn-Shevelev conjecture on a sequence arising in algebraic geometry

I. P. Goulden, S. Litsyn, and V. Shevelev [On a sequence arising in algebraic geometry, J. Integer Sequences 8 (2005), 05.4.7] conjectured that certain Laurent polynomials associated with the solution of a functional equation have only odd negative powers. We prove their conjecture and generalize it.