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

Author

Olivier Gérard

Keywords

• nonn

Links

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

Crossrefs

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

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

Programs

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*)

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