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Generating function in Maple format
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First 15 coefficients

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Av(123, 231) Av(123, 132, 3412) Av(123, 132, 4231) Av(132, 213, 3412) Av(132, 213, 3421) Av(132, 213, 4231) Av(132, 213, 4321) Av(132, 231, 1234) Av(132, 231, 2134) Av(132, 231, 3124) Av(132, 231, 3214)