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A035653
Number of partitions of n into parts 7k and 7k+3 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, 3, 0, 1, 3, 0, 1, 3, 6, 1, 3, 7, 1, 3, 7, 12, 3, 7, 15, 3, 7, 16, 21, 7, 16, 28, 7, 16, 31, 36, 16, 32, 50, 16, 32, 57, 60, 32, 60, 85, 32, 61, 100, 98, 61, 107, 141, 61, 110, 169, 157, 111, 184, 226, 111, 191, 276, 249, 194, 305
OFFSET
1,17
LINKS
FORMULA
G.f. : (-1 + 1/Product_{k>=0} (1 - x^(7 k + 3)))*(-1 + 1/Product_{k>=1} (1 - x^(7 k))). - Robert Price, Aug 12 2020
MATHEMATICA
nmax = 75; s1 = Range[1, nmax/7]*7; s2 = Range[0, nmax/7]*7 + 3;
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^(7 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(7 k + 3)), {k, 0, nmax}]), {x, 0, nmax}], x] (* Robert Price, Aug 12 2020 *)
KEYWORD
nonn
STATUS
approved