A035619 Number of partitions of n into parts 3k and 3k+2 with at least one part of each type.

0, 0, 0, 0, 1, 0, 1, 3, 1, 3, 7, 3, 8, 14, 8, 17, 26, 18, 33, 47, 36, 61, 81, 68, 106, 137, 121, 181, 224, 209, 296, 362, 347, 478, 570, 565, 750, 890, 894, 1166, 1360, 1396, 1774, 2062, 2134, 2677, 3076, 3228, 3973, 4555, 4804, 5854, 6657, 7085, 8513

Offset

1,8

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 80 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 + 2))). - Robert Price, Aug 16 2020

Mathematica

nmax = 55; s1 = Range[1, nmax/3]*3; s2 = Range[0, nmax/3]*3 + 2;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 06 2020 *)
nmax = 55; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(3 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(3 k + 2)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020*)

Crossrefs

Cf. A035441-A035468, A035618, A035620-A035699.

Sequence in context: A356207 A114972 A107461 * A280995 A092689 A281553

Adjacent sequences: A035616 A035617 A035618 * A035620 A035621 A035622

Keyword

nonn

Author

Olivier Gérard

Status

approved