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A035676
Number of partitions of n into parts 8k and 8k+5 with at least one part of each type.
4
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 3, 0, 1, 0, 0, 3, 0, 1, 6, 0, 3, 0, 1, 7, 0, 3, 11, 1, 7, 0, 3, 14, 1, 7, 18, 3, 15, 1, 7, 25, 3, 15, 30, 7, 28, 3, 15, 44, 7, 29, 47, 15, 51, 7, 29, 72, 15, 54, 73, 29, 87, 15, 55, 116, 29, 94, 111, 55, 144, 29, 97, 180, 55
OFFSET
1,21
LINKS
FORMULA
G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8*k + 5)))*(-1 + 1/Product_{k>=1} (1 - x^(8*k))). - Robert Price, Aug 13 2020
MATHEMATICA
nmax = 83; s1 = Range[1, nmax/8]*8; 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 13 2020 *)
nmax = 83; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 5)), {k, 0, nmax}]), {x, 0, nmax}], x] (* Robert Price, Aug 13 2020 *)
KEYWORD
nonn
STATUS
approved