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A035693
Number of partitions of n into parts 8k+3 and 8k+7 with at least one part of each type.
3
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 2, 1, 1, 2, 1, 1, 3, 3, 4, 3, 3, 5, 3, 4, 7, 7, 8, 7, 8, 10, 8, 10, 15, 14, 15, 16, 16, 20, 18, 21, 29, 27, 28, 32, 33, 36, 37, 42, 51, 50, 52, 59, 60, 66, 69, 77, 90, 88, 93, 105, 107, 115, 124, 135, 152, 153, 160, 180, 183, 195
OFFSET
1,18
LINKS
FORMULA
G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8 k + 3)))*(-1 + 1/Product_{k>=0} (1 - x^(8 k + 7))). - Robert Price, Aug 15 2020
MATHEMATICA
nmax = 77; s1 = Range[0, nmax/8]*8 + 3; s2 = Range[0, nmax/8]*8 + 7;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 15 2020 *)
nmax = 77; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 3)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 7)), {k, 0, nmax}]), {x, 0, nmax}], x] (* Robert Price, Aug 15 2020*)
KEYWORD
nonn
STATUS
approved