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A035675
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Number of partitions of n into parts 8k and 8k+4 with at least one part of each type.
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3
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0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 10, 0, 0, 0, 11, 0, 0, 0, 22, 0, 0, 0, 25, 0, 0, 0, 44, 0, 0, 0, 51, 0, 0, 0, 83, 0, 0, 0, 98, 0, 0, 0, 149, 0, 0, 0, 177, 0, 0, 0, 259, 0, 0, 0, 309, 0, 0, 0, 436, 0, 0, 0, 521, 0, 0, 0, 716, 0, 0, 0, 857, 0, 0
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OFFSET
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1,20
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LINKS
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FORMULA
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G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8*k + 4)))*(-1 + 1/Product_{k>=1} (1 - x^(8*k))). - Robert Price, Aug 13 2020
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MATHEMATICA
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nmax = 90; s1 = Range[1, nmax/8]*8; s2 = Range[0, nmax/8]*8 + 4;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 13 2020 *)
nmax = 90; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 4)), {k, 0, nmax}]), {x, 0, nmax}], x] (* Robert Price, Aug 13 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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