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

1,7

Links

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

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

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*)

Crossrefs

Cf. A035441-A035468, A035619-A035699.

Sequence in context: A220010 A190717 A220204 * A220931 A219578 A219373

Adjacent sequences: A035615 A035616 A035617 * A035619 A035620 A035621

Keyword

nonn

Author

Olivier Gérard

Status

approved