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A035694
Number of partitions of n into parts 8k+4 and 8k+5 with at least one part of each type.
4
0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 3, 1, 1, 0, 3, 3, 1, 1, 6, 3, 3, 1, 8, 7, 3, 3, 12, 9, 7, 3, 16, 15, 9, 7, 22, 19, 16, 9, 30, 29, 20, 16, 40, 38, 32, 20, 54, 54, 41, 33, 69, 70, 61, 42, 93, 95, 78, 64, 118, 124, 110, 81, 157, 163, 141, 117, 196, 211, 192, 149, 258
OFFSET
1,17
LINKS
FORMULA
G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8 k + 4)))*(-1 + 1/Product_{k>=0} (1 - x^(8 k + 5))). - Robert Price, Aug 16 2020
MATHEMATICA
nmax = 77; s1 = Range[0, nmax/8]*8 + 4; s2 = Range[0, nmax/8]*8 + 5;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 16 2020 *)
nmax = 77; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 4)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 5)), {k, 0, nmax}]), {x, 0, nmax}], x] (* Robert Price, Aug 16 2020*)
KEYWORD
nonn
STATUS
approved