A035643 Number of partitions of n into parts 6k+1 and 6k+3 with at least one part of each type.

0, 0, 0, 1, 1, 1, 2, 2, 2, 5, 5, 5, 8, 8, 8, 14, 15, 15, 22, 23, 23, 34, 37, 38, 51, 54, 55, 74, 81, 84, 108, 116, 119, 151, 165, 172, 213, 230, 238, 290, 317, 332, 399, 433, 451, 535, 583, 613, 720, 781, 818, 950, 1033, 1088, 1257, 1363, 1432, 1638, 1777, 1875

Offset

1,7

Links

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

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(6 k + 1)))*(-1 + 1/Product_{k>=0} (1 - x^(6 k + 3))). - Robert Price, Aug 16 2020

Mathematica

nmax = 60; s1 = Range[0, nmax/6]*6 + 1; s2 = Range[0, nmax/6]*6 + 3;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 13 2020 *)
nmax = 60; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(6 k + 1)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(6 k + 3)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020 *)

Crossrefs

Cf. A035441-A035468, A035618-A035642, A035644-A035699.

Sequence in context: A020917 A308772 A332966 * A324925 A163946 A240500

Adjacent sequences: A035640 A035641 A035642 * A035644 A035645 A035646

Keyword

nonn

Author

Olivier Gérard

Status

approved