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A035665
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Number of partitions of n into parts 7k+2 and 7k+6 with at least one part of each type.
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3
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0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 2, 2, 2, 2, 2, 3, 4, 6, 4, 7, 4, 8, 6, 12, 10, 13, 12, 14, 15, 18, 21, 24, 23, 29, 26, 35, 33, 46, 41, 52, 50, 58, 61, 71, 77, 84, 89, 99, 101, 120, 121, 146, 142, 168, 166, 192, 199, 226, 239, 259, 275, 300, 316, 354, 370, 416, 422
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OFFSET
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1,14
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LINKS
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FORMULA
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G.f.: (-1 + 1/Product_{k>=0} (1 - x^(7*k + 2)))*(-1 + 1/Product_{k>=0} (1 - x^(7*k + 6))). - Robert Price, Aug 16 2020
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MAPLE
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b:= proc(n, i, t, s) option remember; `if`(n=0, t*s, `if`(i<1, 0,
b(n, i-1, t, s)+(h-> `if`(h in {2, 6}, add(b(n-i*j, i-1,
`if`(h=2, 1, t), `if`(h=6, 1, s)), j=1..n/i), 0))(irem(i, 7))))
end:
a:= n-> b(n$2, 0$2):
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MATHEMATICA
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nmax = 71; s1 = Range[0, nmax/7]*7 + 2; s2 = Range[0, nmax/7]*7 + 6;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 14 2020 *)
nmax = 71; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(7 k + 2)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(7 k + 6)), {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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