A035656 Number of partitions of n into parts 7k and 7k+6 with at least one part of each type.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 1, 3, 6, 0, 0, 0, 1, 3, 7, 11, 0, 0, 1, 3, 7, 14, 18, 0, 1, 3, 7, 15, 25, 29, 1, 3, 7, 15, 28, 43, 45, 3, 7, 15, 29, 50, 70, 69, 7, 15, 29, 53, 85, 112, 103, 15, 29, 54, 92, 140, 172, 153, 29, 54, 95, 155, 222

Offset

1,20

Links

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

Formula

G.f. : (-1 + 1/Product_ {k >= 0} (1 - x^(7 k + 6)))*(-1 + 1/Product_ {k >= 1} (1 - x^(7 k))). - Robert Price, Aug 12 2020

Mathematica

nmax = 81; s1 = Range[1, nmax/7]*7; s2 = Range[0, nmax/7]*7 + 6;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 12 2020 *)
nmax = 81; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(7 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(7 k + 6)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 12 2020 *)

Crossrefs

Cf. A035441-A035468, A035618-A035655, A035657-A035699.

Sequence in context: A007271 A327449 A320179 * A325675 A279948 A264009

Adjacent sequences: A035653 A035654 A035655 * A035657 A035658 A035659

Keyword

nonn

Author

Olivier Gérard

Status

approved