A035694 Number of partitions of n into parts 8k+4 and 8k+5 with at least one part of each type.

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 3, 1, 1, 0, 3, 3, 1, 1, 6, 3, 3, 1, 8, 7, 3, 3, 12, 9, 7, 3, 16, 15, 9, 7, 22, 19, 16, 9, 30, 29, 20, 16, 40, 38, 32, 20, 54, 54, 41, 33, 69, 70, 61, 42, 93, 95, 78, 64, 118, 124, 110, 81, 157, 163, 141, 117, 196, 211, 192, 149, 258

Offset

1,17

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>=0} (1 - x^(8 k + 5))). - Robert Price, Aug 16 2020

Mathematica

nmax = 77; s1 = Range[0, nmax/8]*8 + 4; s2 = Range[0, nmax/8]*8 + 5;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 16 2020 *)
nmax = 77; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 4)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 5)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020*)

Crossrefs

Cf. A035441-A035468, A035618-A035693, A035695-A035699.

Sequence in context: A341412 A196057 A058395 * A006941 A337525 A076277

Adjacent sequences: A035691 A035692 A035693 * A035695 A035696 A035697

Keyword

nonn

Author

Olivier Gérard

Status

approved