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A035655
Number of partitions of n into parts 7k and 7k+5 with at least one part of each type.
3
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 3, 0, 0, 1, 0, 3, 0, 6, 1, 0, 3, 0, 7, 1, 11, 3, 0, 7, 1, 14, 3, 18, 7, 1, 15, 3, 25, 7, 30, 15, 3, 28, 7, 44, 15, 47, 29, 7, 51, 15, 72, 29, 73, 54, 15, 87, 29, 116, 55, 111, 94, 29, 144, 55, 180, 97, 167, 159, 55, 230, 98, 276
OFFSET
1,19
LINKS
FORMULA
G.f. : (-1 + 1/Product_{k>=0} (1 - x^(7 k + 5)))*(-1 + 1/Product_{k>=1} (1 - x^(7 k))). - Robert Price, Aug 12 2020
MATHEMATICA
nmax = 80; s1 = Range[1, nmax/7]*7; 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 12 2020 *)
nmax = 80; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(7 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(7 k + 5)), {k, 0, nmax}]), {x, 0, nmax}], x] (* Robert Price, Aug 12 2020 *)
KEYWORD
nonn
STATUS
approved