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The Harer-Zagier formula provides a three term recurrence relation for the expected value of the single-trace operator $\mathrm {Tr}(X^k)$ where $X$ is a $N\times N$ matrix from the GUE. Is there an analogous recurrence relation for the moments of the GOE?

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It's been a while since the question was asked, but I randomly came across this article by M. Ledoux, which seems to give a five term recurrence relation. See Theorem 2 of:

http://www.numdam.org/article/AIHPB_2009__45_3_754_0.pdf

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