A035675 Number of partitions of n into parts 8k and 8k+4 with at least one part of each type.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 10, 0, 0, 0, 11, 0, 0, 0, 22, 0, 0, 0, 25, 0, 0, 0, 44, 0, 0, 0, 51, 0, 0, 0, 83, 0, 0, 0, 98, 0, 0, 0, 149, 0, 0, 0, 177, 0, 0, 0, 259, 0, 0, 0, 309, 0, 0, 0, 436, 0, 0, 0, 521, 0, 0, 0, 716, 0, 0, 0, 857, 0, 0

Offset

1,20

Links

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

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8*k + 4)))*(-1 + 1/Product_{k>=1} (1 - x^(8*k))). - Robert Price, Aug 13 2020

Mathematica

nmax = 90; s1 = Range[1, nmax/8]*8; 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 13 2020 *)
nmax = 90; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 4)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 13 2020 *)

Crossrefs

Cf. A035441-A035468, A035618-A035674, A035676-A035699.

Sequence in context: A028622 A028614 A028703 * A356500 A028606 A284445

Adjacent sequences: A035672 A035673 A035674 * A035676 A035677 A035678

Keyword

nonn

Author

Olivier Gérard

Status

approved