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

Author

Olivier Gérard

Keywords

• nonn

Links

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

Crossrefs

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

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

Programs

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 *)

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