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A035643
Number of partitions of n into parts 6k+1 and 6k+3 with at least one part of each type.
3
0, 0, 0, 1, 1, 1, 2, 2, 2, 5, 5, 5, 8, 8, 8, 14, 15, 15, 22, 23, 23, 34, 37, 38, 51, 54, 55, 74, 81, 84, 108, 116, 119, 151, 165, 172, 213, 230, 238, 290, 317, 332, 399, 433, 451, 535, 583, 613, 720, 781, 818, 950, 1033, 1088, 1257, 1363, 1432, 1638, 1777, 1875
OFFSET
1,7
LINKS
FORMULA
G.f.: (-1 + 1/Product_{k>=0} (1 - x^(6 k + 1)))*(-1 + 1/Product_{k>=0} (1 - x^(6 k + 3))). - Robert Price, Aug 16 2020
MATHEMATICA
nmax = 60; s1 = Range[0, nmax/6]*6 + 1; s2 = Range[0, nmax/6]*6 + 3;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 13 2020 *)
nmax = 60; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(6 k + 1)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(6 k + 3)), {k, 0, nmax}]), {x, 0, nmax}], x] (* Robert Price, Aug 16 2020 *)
KEYWORD
nonn
STATUS
approved