A035618 Number of partitions of n into parts 3k and 3k+1 with at least one part of each type.

0, 0, 0, 1, 1, 1, 4, 4, 4, 10, 11, 11, 22, 25, 26, 44, 51, 54, 84, 98, 105, 152, 178, 193, 266, 312, 341, 452, 528, 581, 749, 873, 964, 1214, 1409, 1561, 1930, 2234, 2479, 3018, 3478, 3866, 4647, 5339, 5937, 7061, 8081, 8991, 10594, 12089, 13447, 15721

Offset: 1

Author

Olivier Gérard

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 75 terms from Robert Price)

Crossrefs

Cf. A035441-A035468, A035619-A035699.

Programs

Mathematica

nmax = 52; kmax = nmax/3; s1 = Range[1, nmax/3]*3; s2 = Range[0, nmax/3]*3 + 1;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 06 2020 *)
nmax = 52; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(3 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(3 k + 1)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020*)

Formula

G.f.: (-1 + 1/Product_{k>=1} (1 - x^(3 k)))*(-1 + 1/Product_{k>=0} (1 - x^(3 k + 1))). - Robert Price, Aug 16 2020