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A361655
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Number of even-length integer partitions of 2n with integer mean.
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3
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0, 1, 3, 4, 10, 6, 33, 8, 65, 68, 117, 12, 583, 14, 319, 1078, 1416, 18, 3341, 20, 8035, 5799, 1657, 24, 36708, 16954, 3496, 24553, 68528, 30, 192180, 32, 178802, 91561, 14625, 485598, 955142, 38, 29223, 316085, 2622697, 42, 3528870, 44, 2443527, 5740043
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OFFSET
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0,3
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LINKS
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EXAMPLE
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The a(0) = 0 through a(5) = 6 partitions:
. (11) (22) (33) (44) (55)
(31) (42) (53) (64)
(1111) (51) (62) (73)
(111111) (71) (82)
(2222) (91)
(3221) (1111111111)
(3311)
(4211)
(5111)
(11111111)
For example, the partition (4,2,1,1) has length 4 and mean 2, so is counted under a(4).
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[2n], EvenQ[Length[#]]&&IntegerQ[Mean[#]]&]], {n, 0, 15}]
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PROG
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(PARI) a(n)=if(n==0, 0, sumdiv(n, d, polcoef(1/prod(k=1, 2*d, 1 - x^k + O(x*x^(2*(n-d)))), 2*(n-d)))) \\ Andrew Howroyd, Mar 24 2023
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CROSSREFS
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Including odd-length partitions gives A067538 bisected, ranks A316413.
For median instead of mean we have A361653.
The odd-length version is counted by A361656.
A326622 counts factorizations with integer mean.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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