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A035625
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Number of partitions of n into parts 4k+1 and 4k+3 with at least one part of each type.
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3
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0, 0, 0, 1, 1, 1, 2, 4, 4, 5, 7, 10, 12, 14, 18, 24, 28, 33, 41, 50, 59, 70, 84, 100, 117, 137, 161, 188, 219, 254, 295, 341, 393, 453, 520, 595, 682, 780, 889, 1011, 1150, 1307, 1481, 1673, 1893, 2140, 2411, 2713, 3053, 3433, 3851, 4313, 4833, 5411
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OFFSET
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1,7
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LINKS
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FORMULA
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G.f.: (-1 + 1/Product_{k>=0} (1 - x^(4 k + 1)))*(-1 + 1/Product_{k>=0} (1 - x^(4 k + 3))). - Robert Price, Aug 16 2020
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MATHEMATICA
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nmax = 54; s1 = Range[0, nmax/4]*4 + 1; s2 = Range[0, nmax/4]*4 + 3;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 06 2020 *)
nmax = 54; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(4 k + 3)), {k, 0, 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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