A035690 Number of partitions of n into parts 8k+3 and 8k+4 with at least one part of each type.

0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 3, 1, 1, 3, 4, 1, 3, 4, 7, 3, 4, 8, 10, 4, 8, 11, 15, 8, 11, 18, 21, 11, 19, 24, 30, 19, 25, 37, 42, 25, 40, 50, 56, 41, 53, 70, 79, 54, 77, 95, 103, 80, 103, 129, 141, 106, 144, 172, 183, 151, 189, 228, 246, 197, 257, 301, 314

Offset

1,15

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 + 4))). - Robert Price, Aug 15 2020

Mathematica

nmax = 71; s1 = Range[0, nmax/8]*8 + 3; s2 = Range[0, nmax/8]*8 + 4;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 15 2020 *)
nmax = 71; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 3)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 4)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 15 2020*)

Crossrefs

Cf. A035441-A035468, A035618-A035689, A035691-A035699.

Sequence in context: A114476 A260419 A117184 * A124794 A206496 A097560

Adjacent sequences: A035687 A035688 A035689 * A035691 A035692 A035693

Keyword

nonn

Author

Olivier Gérard

Status

approved