A035654 Number of partitions of n into parts 7k and 7k+4 with at least one part of each type.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 3, 1, 0, 0, 3, 1, 0, 6, 3, 1, 0, 7, 3, 1, 11, 7, 3, 1, 14, 7, 3, 19, 15, 7, 3, 26, 15, 7, 32, 29, 15, 7, 46, 30, 15, 51, 53, 30, 15, 76, 56, 30, 81, 91, 57, 30, 124, 98, 57, 126, 152, 101, 57, 195, 167, 102, 195, 245, 174, 102, 304

Offset

1,18

Links

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

Formula

G.f. : (-1 + 1/Product_{k>=0} (1 - x^(7 k + 4)))*(-1 + 1/Product_{k>=1} (1 - x^(7 k))). - Robert Price, Aug 12 2020

Mathematica

nmax = 78; s1 = Range[1, nmax/7]*7; 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 12 2020 *)
nmax = 78; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(7 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(7 k + 4)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 12 2020 *)

Crossrefs

Cf. A035441-A035468, A035618-A035653, A035655-A035699.

Sequence in context: A065413 A107131 A027200 * A170846 A085604 A345371

Adjacent sequences: A035651 A035652 A035653 * A035655 A035656 A035657

Keyword

nonn

Author

Olivier Gérard

Status

approved