A035674 Number of partitions of n into parts 8k and 8k+3 with at least one part of each type.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 3, 1, 0, 3, 1, 0, 3, 1, 6, 3, 1, 7, 3, 1, 7, 3, 12, 7, 3, 15, 7, 3, 16, 7, 21, 16, 7, 28, 16, 7, 31, 16, 36, 32, 16, 50, 32, 16, 57, 32, 60, 60, 32, 85, 61, 32, 100, 61, 98, 107, 61, 141, 110, 61, 169, 111, 157, 184, 111, 226

Offset

1,19

Links

Robert Price, Table of n, a(n) for n = 1..1000

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8*k + 3)))*(-1 + 1/Product_{k>=1} (1 - x^(8*k))). - Robert Price, Aug 12 2020

Mathematica

nmax = 78; s1 = Range[1, nmax/8]*8; s2 = Range[0, nmax/8]*8 + 3;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 12 2020 *)
nmax = 78; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 3)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 12 2020 *)

Crossrefs

Cf. A035441-A035468, A035618-A035673, A035675-A035699.

Sequence in context: A163972 A068464 A244679 * A058600 A133704 A160019

Adjacent sequences: A035671 A035672 A035673 * A035675 A035676 A035677

Keyword

nonn

Author

Olivier Gérard

Status

approved