A035639 Number of partitions of n into parts 6k and 6k+3 with at least one part of each type.

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 4, 0, 0, 4, 0, 0, 10, 0, 0, 11, 0, 0, 22, 0, 0, 25, 0, 0, 44, 0, 0, 51, 0, 0, 83, 0, 0, 98, 0, 0, 149, 0, 0, 177, 0, 0, 259, 0, 0, 309, 0, 0, 436, 0, 0, 521, 0, 0, 716, 0, 0, 857, 0, 0, 1151, 0, 0, 1376, 0, 0, 1816, 0, 0, 2170, 0, 0, 2818, 0, 0

Offset

1,15

Links

Robert Price, Table of n, a(n) for n = 1..999

Formula

G.f. : (-1 + 1/Product_{k>=0} (1 - x^(6 k + 3)))*(-1 + 1/Product_{k>=1} (1 - x^(6 k))). - Robert Price, Aug 12 2020

Mathematica

nmax = 83; s1 = Range[1, nmax/6]*6; 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 12 2020 *)
nmax = 83; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(6 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(6 k + 3)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 12 2020 *)

Crossrefs

Cf. A035441-A035468, A035618-A035638, A035640-A035699.

First trisection gives A006477.

Sequence in context: A308277 A278216 A036480 * A284689 A037214 A245198

Adjacent sequences: A035636 A035637 A035638 * A035640 A035641 A035642

Keyword

nonn

Author

Olivier Gérard

Status

approved