A035671 Number of partitions of n into parts 7k+5 and 7k+6 with at least one part of each type.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 2, 0, 0, 1, 1, 3, 2, 3, 1, 1, 3, 3, 6, 5, 5, 3, 3, 7, 8, 11, 9, 8, 7, 9, 15, 15, 19, 16, 15, 16, 19, 27, 28, 32, 28, 27, 32, 36, 48, 48, 52, 49, 49, 58, 65, 80, 81, 85, 84, 84, 101, 111, 131, 132, 137, 138, 143, 169, 184, 208, 213

Offset: 1

Author

Olivier Gérard

Keywords

• nonn

Links

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

Crossrefs

Cf. A035441-A035468, A035618-A035670, A035672-A035699.

Programs

Mathematica

nmax = 78; s1 = Range[0, nmax/7]*7 + 5; s2 = Range[0, nmax/7]*7 + 6;
Table[Count[IntegerPartitions[n, All, s1~Join~s2], x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 15 2020 *)
nmax = 78; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(7 k + 5)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(7 k + 6)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 15 2020 *)

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(7*k + 5)))*(-1 + 1/Product_{k>=0} (1 - x^(7*k + 6))). - Robert Price, Aug 15 2020