A035463 Number of partitions of n into parts 8k+4 or 8k+5.

0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 2, 2, 1, 1, 2, 3, 2, 1, 4, 4, 3, 2, 5, 7, 5, 3, 7, 9, 8, 5, 9, 13, 11, 8, 13, 17, 16, 12, 18, 24, 22, 17, 24, 32, 31, 24, 32, 43, 42, 34, 43, 56, 57, 47, 57, 74, 75, 64, 76, 96, 100, 86, 99, 126, 130, 115, 129, 161, 171, 151, 168, 207, 219, 200

Offset: 1

Author

Olivier Gérard

Keywords

• nonn

Links

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

Crossrefs

Cf. A035694.

Programs

Mathematica

nmax = 100; Rest[CoefficientList[Series[Product[1/((1 - x^(8k+4))*(1 - x^(8k+5))), {k, 0, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 27 2015 *)
nmax = 60; kmax = nmax/8;
s = Flatten[{Range[0, kmax]*8 + 4}~Join~{Range[0, kmax]*8 + 5}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 1, nmax}] (* Robert Price, Aug 04 2020 *)

Formula

a(n) ~ exp(Pi*sqrt(n/6)) * Gamma(5/8) / (4 * 2^(9/16) * 3^(5/16) * Pi^(3/8) * n^(13/16)). - Vaclav Kotesovec, Aug 27 2015