A035691 Number of partitions of n into parts 8k+3 and 8k+5 with at least one part of each type.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 3, 1, 1, 3, 1, 3, 3, 2, 6, 3, 4, 7, 4, 8, 7, 6, 13, 8, 10, 15, 10, 17, 17, 14, 24, 19, 22, 30, 23, 33, 34, 31, 46, 39, 44, 56, 47, 63, 65, 61, 82, 75, 84, 101, 90, 113, 118, 115, 145, 137, 151, 176, 165, 197, 207, 206, 246, 242, 264

Offset

1,16

Links

Alois P. Heinz, Table of n, a(n) for n = 1..5000

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8 k + 3)))*(-1 + 1/Product_{k>=0} (1 - x^(8 k + 5))). - Robert Price, Aug 15 2020

Mathematica

nmax = 74; s1 = Range[0, nmax/8]*8 + 3; s2 = Range[0, nmax/8]*8 + 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^(8 k + 3)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 5)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 15 2020*)

Crossrefs

Cf. A035441-A035468, A035618-A035690, A035692-A035699.

Sequence in context: A131779 A131775 A218825 * A131781 A082465 A226856

Adjacent sequences: A035688 A035689 A035690 * A035692 A035693 A035694

Keyword

nonn

Author

Olivier Gérard

Status

approved