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A035656
Number of partitions of n into parts 7k and 7k+6 with at least one part of each type.
3
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 1, 3, 6, 0, 0, 0, 1, 3, 7, 11, 0, 0, 1, 3, 7, 14, 18, 0, 1, 3, 7, 15, 25, 29, 1, 3, 7, 15, 28, 43, 45, 3, 7, 15, 29, 50, 70, 69, 7, 15, 29, 53, 85, 112, 103, 15, 29, 54, 92, 140, 172, 153, 29, 54, 95, 155, 222
OFFSET
1,20
LINKS
FORMULA
G.f. : (-1 + 1/Product_ {k >= 0} (1 - x^(7 k + 6)))*(-1 + 1/Product_ {k >= 1} (1 - x^(7 k))). - Robert Price, Aug 12 2020
MATHEMATICA
nmax = 81; s1 = Range[1, nmax/7]*7; 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 12 2020 *)
nmax = 81; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(7 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(7 k + 6)), {k, 0, nmax}]), {x, 0, nmax}], x] (* Robert Price, Aug 12 2020 *)
KEYWORD
nonn
STATUS
approved