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A304620
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Expansion of (1/(1 - x)) * Sum_{k>=0} x^(2*k) / Product_{j=1..2*k} (1 - x^j).
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7
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1, 1, 2, 3, 6, 9, 15, 22, 34, 48, 70, 97, 137, 186, 255, 341, 459, 605, 800, 1042, 1359, 1751, 2256, 2879, 3672, 4645, 5869, 7367, 9234, 11508, 14319, 17730, 21916, 26975, 33143, 40570, 49575, 60376, 73402, 88974, 107666, 129933, 156546, 188148, 225767, 270300, 323115, 385453
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OFFSET
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0,3
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COMMENTS
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Also the number of integer partitions of 2n+1 with odd greatest part and alternating sum 1, where the alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i, which is equal to the number of odd parts in the conjugate partition. For example, the a(0) = 1 through a(6) = 15 partitions are:
1 111 32 331 54 551 76
11111 3211 3222 3332 5422
1111111 3321 5411 5521
33111 33221 33331
321111 322211 55111
111111111 332111 322222
3311111 332221
32111111 333211
11111111111 541111
3322111
32221111
33211111
331111111
3211111111
1111111111111
Also odd-length partitions of 2n+1 with exactly one odd part.
(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)/Product[(1 - x^j), {j, 1, 2 k}], {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], OddQ[Length[#]]&&Count[#, _?OddQ]==1&]], {n, 1, 30, 2}] (* Gus Wiseman, Jun 26 2021 *)
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CROSSREFS
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The version for even instead of odd greatest part is A306145.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A000070 counts partitions with alternating sum 1.
A067661 counts strict partitions of even length.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A344610 counts partitions by sum and positive reverse-alternating sum.
Cf. A000097, A006330, A027193, A030229, A067659, A236559, A236914, A239829, A239830, A318156, 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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