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

1,18

Links

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

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

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 *)

Crossrefs

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

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

Sequence in context: A019920 A246130 A010675 * A035638 A347156 A292142

Adjacent sequences: A035670 A035671 A035672 * A035674 A035675 A035676

Keyword

nonn

Author

Olivier Gérard

Status

approved