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COMMENT
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The 77 applicable partitions range from n = 20 through n=26 with the following frequency distribution: 2, 9, 16, 15, 14, 9 and 12.
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EXAMPLE
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A110624(1) = 2 because the two relevant partitions of 20 are 4+4+4+4+4 and 5+5+5+5 and can not be permuted.
A110624(2) = 148 because at n = 21 the relevant partitions can be permuted in 20 plus 128 = 148 ways.
The 20 ways are apparent from A034261 by observing that 9 + 11 = 20 in the subsequences 1,9,39,... and 1,11,56,...
and the 128 ways result from examining the nine partitions of n = 21 having 3,31,32,33,7,211,221,311 and 411 degrees.
A110624(4)= A110554(13) = 16746.
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