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A035657
Number of partitions of n into parts 7k+1 and 7k+2 with at least one part of each type.
3
0, 0, 1, 1, 2, 2, 3, 3, 4, 6, 7, 9, 10, 12, 13, 15, 19, 22, 27, 30, 35, 38, 43, 50, 57, 67, 74, 85, 92, 103, 115, 130, 148, 165, 185, 202, 223, 246, 273, 306, 337, 376, 408, 449, 488, 539, 594, 654, 720, 784, 855, 928, 1013, 1109, 1210, 1326, 1436, 1563, 1685
OFFSET
1,5
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 125 terms from Robert Price)
FORMULA
G.f.: (-1 + 1/Product_{k>=0} (1 - x^(7 k + 1)))*(-1 + 1/Product_{k>=0} (1 - x^(7 k + 2))). - Robert Price, Aug 16 2020
MAPLE
b:= proc(n, i, t, s) option remember; `if`(n=0, t*s, `if`(i<1, 0,
b(n, i-1, t, s)+(h-> `if`(h in {1, 2}, add(b(n-i*j, i-1,
`if`(h=1, 1, t), `if`(h=2, 1, s)), j=1..n/i), 0))(irem(i, 7))))
end:
a:= n-> b(n$2, 0$2):
seq(a(n), n=1..75); # Alois P. Heinz, Aug 14 2020
MATHEMATICA
nmax = 59; s1 = Range[0, nmax/7]*7 + 1; s2 = Range[0, nmax/7]*7 + 2;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 14 2020 *)
nmax = 59; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(7 k + 1)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(7 k + 2)), {k, 0, nmax}]), {x, 0, nmax}], x] (* Robert Price, Aug 16 2020 *)
KEYWORD
nonn
STATUS
approved