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Number of partitions of n into parts 6k and 6k+5 with at least one part of each type.
G.f. : (-1 + 1/Product_ {k >= 0} (1 - x^(6 k + 5)))*(-1 + 1/Product_ {k >= 1} (1 - x^(6 k))). - Robert Price, Aug 12 2020
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Robert Price, <a href="/A035641/b035641.txt">Table of n, a(n) for n = 1..1000</a>
G.f. : (-1 + 1/Product_ {k >= 0} (1 - x^(6 k + 5)))*(-1 + 1/Product_ {k >= 1} (1 - x^(6 k))). - Robert Price, Aug 12 2020
nmax = 75; s1 = Range[1, nmax/6]*6; s2 = Range[0, nmax/6]*6 + 5;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 12 2020 *)
nmax = 75; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(6 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(6 k + 5)), {k, 0, nmax}]), {x, 0, nmax}], x] (* Robert Price, Aug 12 2020 *)
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Olivier Gerard (olivier.gerard(AT)gmail.com)