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A035650
Number of partitions of n into parts 6k+3 and 6k+5 with at least one part of each type.
3
0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 3, 0, 1, 3, 1, 3, 6, 1, 3, 8, 3, 7, 12, 3, 9, 16, 7, 15, 22, 9, 19, 30, 16, 29, 40, 20, 38, 54, 32, 54, 69, 41, 70, 93, 61, 95, 118, 78, 124, 156, 110, 163, 195, 141, 211, 255, 192, 271, 317, 245, 349, 409, 323, 441, 506, 412, 562, 644
OFFSET
1,14
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^(6 k + 3)))*(-1 + 1/Product_{k>=0} (1 - x^(6 k + 6))). - 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 {3, 5}, add(b(n-i*j, i-1,
`if`(h=3, 1, t), `if`(h=5, 1, s)), j=1..n/i), 0))(irem(i, 6))))
end:
a:= n-> b(n$2, 0$2):
seq(a(n), n=1..75); # Alois P. Heinz, Aug 14 2020
MATHEMATICA
nmax = 71; s1 = Range[0, nmax/6]*6 + 3; s2 = Range[0, nmax/6]*6 + 5;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 14 2020 *)
nmax = 71; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(6 k + 3)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(6 k + 5)), {k, 0, nmax}]), {x, 0, nmax}], x] (* Robert Price, Aug 16 2020 *)
KEYWORD
nonn
STATUS
approved