A035673 Number of partitions of n into parts 8k and 8k+2 with at least one part of each type.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 4, 0, 4, 0, 4, 0, 4, 0, 10, 0, 11, 0, 11, 0, 11, 0, 22, 0, 25, 0, 26, 0, 26, 0, 44, 0, 51, 0, 54, 0, 55, 0, 84, 0, 98, 0, 105, 0, 108, 0, 153, 0, 178, 0, 193, 0, 200, 0, 269, 0, 313, 0, 341, 0, 356, 0, 459, 0, 531, 0, 582, 0, 611, 0

Offset: 1

Author

Olivier Gérard

Keywords

• nonn

Links

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

Crossrefs

Cf. A035441-A035468, A035618-A035672, A035674-A035699.

Bisections give: A035621 (even part), A000004 (odd part).

Programs

Mathematica

nmax = 81; s1 = Range[1, nmax/8]*8; s2 = Range[0, nmax/8]*8 + 2;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 12 2020 *)
nmax = 81; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 2)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 12 2020 *)

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8*k + 2)))*(-1 + 1/Product_{k>=1} (1 - x^(8*k))). - Robert Price, Aug 12 2020