A035669 Number of partitions of n into parts 7k+4 and 7k+5 with at least one part of each type.

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 2, 1, 1, 1, 2, 3, 1, 4, 3, 3, 3, 5, 7, 4, 7, 7, 8, 8, 10, 13, 10, 14, 15, 16, 17, 20, 25, 21, 26, 29, 32, 33, 37, 45, 41, 47, 54, 58, 61, 65, 79, 76, 83, 94, 103, 108, 113, 132, 135, 143, 160, 172, 185, 192, 219, 227, 240, 265, 286

Offset

1,16

Links

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

Formula

G.f. : (-1 + 1/Product_{k>=0} (1 - x^(7 k + 4)))*(-1 + 1/Product_{k>=0} (1 - x^(7 k + 5)). - Robert Price, Aug 15 2020

Mathematica

nmax = 74; s1 = Range[0, nmax/7]*7 + 4; s2 = Range[0, nmax/7]*7 + 5;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 15 2020 *)
nmax = 74; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(7 k + 4)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(7 k + 5)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 15 2020*)

Crossrefs

Cf. A035441-A035468, A035618-A035668, A035670-A035699.

Sequence in context: A249298 A088863 A053283 * A126863 A106806 A178526

Adjacent sequences: A035666 A035667 A035668 * A035670 A035671 A035672

Keyword

nonn

Author

Olivier Gérard

Status

approved