A035695 Number of partitions of n into parts 8k+4 and 8k+6 with at least one part of each type.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 3, 0, 1, 0, 4, 0, 3, 0, 7, 0, 4, 0, 10, 0, 8, 0, 15, 0, 11, 0, 21, 0, 18, 0, 30, 0, 24, 0, 42, 0, 37, 0, 56, 0, 50, 0, 78, 0, 70, 0, 102, 0, 95, 0, 137, 0, 129, 0, 179, 0, 171, 0, 236, 0, 227, 0, 303, 0, 297, 0, 395, 0, 386, 0, 502, 0

Offset

1,18

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8 k + 4)))*(-1 + 1/Product_{k>=0} (1 - x^(8 k + 6))). - Robert Price, Aug 16 2020

Mathematica

nmax = 83; s1 = Range[0, nmax/8]*8 + 4; s2 = Range[0, nmax/8]*8 + 6;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 16 2020 *)
nmax = 83; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 4)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 6)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020*)

Crossrefs

Bisections give A035626 (even part), A000004 (odd part).

Cf. A035441-A035468, A035618-A035694, A035696-A035699.

Sequence in context: A111526 A117178 A111527 * A100257 A318315 A329861

Adjacent sequences: A035692 A035693 A035694 * A035696 A035697 A035698

Keyword

nonn

Author

Olivier Gérard

Status

approved