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A306145
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Expansion of (1/(1 - x)) * Sum_{k>=0} x^(2*k+1) / Product_{j=1..2*k+1} (1 - x^j).
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6
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0, 1, 2, 4, 6, 10, 15, 23, 33, 49, 69, 98, 135, 187, 253, 343, 456, 607, 797, 1045, 1355, 1755, 2252, 2884, 3666, 4651, 5863, 7375, 9226, 11517, 14310, 17741, 21904, 26988, 33130, 40586, 49558, 60394, 73383, 88996, 107642, 129958, 156519, 188178, 225734, 270335, 323078, 385494
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OFFSET
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0,3
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COMMENTS
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Also the number of even-length integer partitions of 2n+1 with exactly one odd part. For example, the a(1) = 1 through a(5) = 10 partitions are:
(2,1) (3,2) (4,3) (5,4) (6,5)
(4,1) (5,2) (6,3) (7,4)
(6,1) (7,2) (8,3)
(2,2,2,1) (8,1) (9,2)
(3,2,2,2) (10,1)
(4,2,2,1) (4,3,2,2)
(4,4,2,1)
(5,2,2,2)
(6,2,2,1)
(2,2,2,2,2,1)
Also partitions of 2n+1 with even greatest part and alternating sum 1.
(End)
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LINKS
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FORMULA
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a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(5/2)*Pi*sqrt(n)). - Vaclav Kotesovec, Aug 20 2018
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MATHEMATICA
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nmax = 47; CoefficientList[Series[1/(1 - x) Sum[x^(2 k + 1)/Product[(1 - x^j), {j, 1, 2 k + 1}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 47; CoefficientList[Series[(1 - EllipticTheta[4, 0, x])/(2 (1 - x) QPochhammer[x]), {x, 0, nmax}], x]
Table[Length[Select[IntegerPartitions[n], EvenQ[Length[#]]&&Count[#, _?OddQ]==1&]], {n, 1, 30, 2}] (* Gus Wiseman, Jun 23 2021 *)
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CROSSREFS
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The ordered version appears to be A087447 modulo initial terms.
The version for odd instead of even-length partitions is A304620.
The case of strict partitions is A318156.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A344610 counts partitions by sum and positive reverse-alternating sum.
Cf. A000070, A000097, A006330, A030229, A067659, A236559, A236914, A239829, A239830, A338907, A344611.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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