A035640 Number of partitions of n into parts 6k and 6k+4 with at least one part of each type.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 3, 0, 1, 0, 3, 0, 7, 0, 3, 0, 8, 0, 14, 0, 8, 0, 17, 0, 26, 0, 18, 0, 33, 0, 47, 0, 36, 0, 61, 0, 81, 0, 68, 0, 106, 0, 137, 0, 121, 0, 181, 0, 224, 0, 209, 0, 296, 0, 362, 0, 347, 0, 478, 0, 570, 0, 565, 0, 750, 0, 890, 0, 894, 0, 1166, 0

Offset

1,16

Links

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

Formula

G.f. : (-1 + 1/Product_{k>=0} (1 - x^(6 k + 4)))*(-1 + 1/Product_{k>=1} (1 - x^(6 k))). - Robert Price, Aug 12 2020

Mathematica

nmax = 81; s1 = Range[1, nmax/6]*6; s2 = Range[0, nmax/6]*6 + 4;
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^(6 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(6 k + 4)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 12 2020 *)

Crossrefs

Cf. A035441-A035468, A035618-A035639, A035641-A035699.

Bisections give: A035619 (even part), A000004 (odd part).

Sequence in context: A053385 A213543 A247254 * A079327 A340504 A242447

Adjacent sequences: A035637 A035638 A035639 * A035641 A035642 A035643

Keyword

nonn

Author

Olivier Gérard

Status

approved