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Generating function in Maple format
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First 15 coefficients

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Av(12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 13425, 13452, 13524, 13542, 14235, 14253, 14325, 14352, 14523, 14532, 15234, 15243, 15324, 15342, 15423, 15432, 23415, 23451, 23514, 23541, 24315, 24351, 24513, 24531, 25314, 25341, 25413, 25431, 34512, 34521, 35412, 35421) Av(12345, 12354, 12435, 12453, 12534, 12543, 13425, 13452, 13524, 13542, 14523, 14532, 21345, 21354, 21435, 21453, 21534, 21543, 23415, 23451, 23514, 23541, 24513, 24531, 31425, 31452, 31524, 31542, 32415, 32451, 32514, 32541, 34512, 34521, 41523, 41532, 42513, 42531, 43512, 43521) Av(12345, 12354, 12435, 12453, 12534, 12543, 13452, 13542, 21345, 21354, 21435, 21453, 21534, 21543, 23451, 23541, 31245, 31254, 31425, 31452, 31524, 31542, 32451, 32541, 41235, 41253, 41325, 41352, 41523, 41532, 42351, 42531, 51234, 51243, 51324, 51342, 51423, 51432, 52341, 52431) Av(12354, 12453, 13254, 13452, 14253, 14352, 15243, 15342, 21354, 21453, 23154, 23451, 24153, 24351, 25143, 25341, 31254, 31452, 32154, 32451, 34152, 34251, 35142, 35241, 41253, 41352, 42153, 42351, 43152, 43251, 45132, 45231, 51243, 51342, 52143, 52341, 53142, 53241, 54132, 54231) Av(12435, 12453, 12543, 14235, 14253, 14523, 15243, 15423, 21435, 21453, 21543, 24135, 24153, 24315, 24351, 24513, 24531, 25143, 25413, 25431, 41235, 41253, 41523, 42135, 42153, 42315, 42351, 42513, 42531, 45123, 45213, 45231, 51243, 51423, 52143, 52413, 52431, 54123, 54213, 54231)