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A035667
Number of partitions of n into parts 7k+3 and 7k+5 with at least one part of each type.
3
0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 2, 1, 1, 3, 1, 3, 3, 4, 4, 3, 7, 4, 7, 8, 8, 10, 8, 14, 11, 14, 18, 16, 20, 19, 27, 24, 28, 35, 31, 40, 40, 48, 48, 53, 64, 60, 73, 74, 86, 90, 96, 114, 108, 129, 135, 149, 159, 167, 196, 190, 221, 234, 249, 274, 285, 326, 324, 367
OFFSET
1,15
LINKS
FORMULA
G.f. : (-1 + 1/Product_{k>=0} (1 - x^(7 k + 3)))*(-1 + 1/Product_{k>=0} (1 - x^(7 k + 5)). - Robert Price, Aug 15 2020
MATHEMATICA
nmax = 72; s1 = Range[0, nmax/7]*7 + 3; s2 = Range[0, nmax/7]*7 + 5;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 15 2020 *)
nmax = 72; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(7 k + 3)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(7 k + 5)), {k, 0, nmax}]), {x, 0, nmax}], x] (* Robert Price, Aug 15 2020*)
KEYWORD
nonn
STATUS
approved