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We compute the natural generating function ${\\bf C}_g(z)=\\sum_{n\\geq 0} {\\bf c}_g(n)z^n$ for the number ${\\bf c}_g(n)$ of linear chord diagrams of fixed genus $g\\geq 1$ with a given number $n\\geq 0$ of chords and find the remarkably simple formula ${\\bf C}_g(z)=z^{2g}R_g(z) (1-4z)^{{1\\over 2}-3g}$, where $R_g(z)$ is a polynomial of degree at most $g-1$ with integral coefficients satisfying $R_g({1\\over 4})\\neq 0$ and $R_g(0) = {\\bf c}_g(2g)\\neq 0.$ In particular, ${\\bf C}_g(z)$ is algebraic over $\\m","authors_text":"C. M. Reidys, J. E. Andersen, M. S. Waterman, R. C. 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