A035666 Number of partitions of n into parts 7k+3 and 7k+4 with at least one part of each type.

0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 3, 1, 1, 3, 3, 2, 3, 6, 4, 4, 7, 8, 6, 8, 13, 10, 10, 16, 17, 14, 19, 25, 22, 23, 32, 34, 31, 38, 48, 45, 47, 60, 65, 62, 73, 86, 86, 90, 109, 117, 117, 133, 153, 155, 165, 191, 205, 209, 235, 261, 272, 288, 326, 349, 362, 398, 440, 459

Offset

1,14

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Formula

G.f. : (-1 + 1/Product_{k>=0} (1 - x^(7 k + 3)))*(-1 + 1/Product_{k>=0} (1 - x^(7 k + 4)). - Robert Price, Aug 15 2020

Mathematica

nmax = 71; s1 = Range[0, nmax/7]*7 + 3; s2 = Range[0, nmax/7]*7 + 4;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 15 2020 *)
nmax = 71; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(7 k + 3)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(7 k + 4)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 15 2020*)

Crossrefs

Cf. A035441-A035468, A035618-A035665, A035667-A035699.

Sequence in context: A035649 A278509 A094782 * A060592 A080426 A230293

Adjacent sequences: A035663 A035664 A035665 * A035667 A035668 A035669

Keyword

nonn

Author

Olivier Gérard

Status

approved