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A035654
Number of partitions of n into parts 7k and 7k+4 with at least one part of each type.
3
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 3, 1, 0, 0, 3, 1, 0, 6, 3, 1, 0, 7, 3, 1, 11, 7, 3, 1, 14, 7, 3, 19, 15, 7, 3, 26, 15, 7, 32, 29, 15, 7, 46, 30, 15, 51, 53, 30, 15, 76, 56, 30, 81, 91, 57, 30, 124, 98, 57, 126, 152, 101, 57, 195, 167, 102, 195, 245, 174, 102, 304
OFFSET
1,18
LINKS
FORMULA
G.f. : (-1 + 1/Product_{k>=0} (1 - x^(7 k + 4)))*(-1 + 1/Product_{k>=1} (1 - x^(7 k))). - Robert Price, Aug 12 2020
MATHEMATICA
nmax = 78; s1 = Range[1, nmax/7]*7; s2 = Range[0, nmax/7]*7 + 4;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 12 2020 *)
nmax = 78; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(7 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(7 k + 4)), {k, 0, nmax}]), {x, 0, nmax}], x] (* Robert Price, Aug 12 2020 *)
KEYWORD
nonn
STATUS
approved