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A035621
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Number of partitions of n into parts 4k and 4k+1 with at least one part of each type.
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4
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0, 0, 0, 0, 1, 1, 1, 1, 4, 4, 4, 4, 10, 11, 11, 11, 22, 25, 26, 26, 44, 51, 54, 55, 84, 98, 105, 108, 153, 178, 193, 200, 269, 313, 341, 356, 459, 531, 582, 611, 764, 880, 967, 1021, 1244, 1424, 1568, 1662, 1988, 2264, 2494, 2653, 3122, 3536, 3896, 4155
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OFFSET
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1,9
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LINKS
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FORMULA
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G.f.: (-1 + 1/Product_{k>=1} (1 - x^(4 k)))*(-1 + 1/Product_{k>=0} (1 - x^(4 k + 1))). - Robert Price, Aug 16 2020
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MATHEMATICA
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nmax = 56; s1 = Range[1, nmax/4]*4; s2 = Range[0, nmax/4]*4 + 1;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 06 2020 *)
nmax = 56; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(4 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(4 k + 1)), {k, 0, nmax}]), {x, 0, nmax}], x] (* Robert Price, Aug 16 2020*)
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CROSSREFS
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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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