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

1,18

Links

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

Formula

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

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 *)

Crossrefs

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

Sequence in context: A164925 A393822 A334546 * A190164 A253938 A131488

Adjacent sequences: A035668 A035669 A035670 * A035672 A035673 A035674

Keyword

nonn

Author

Olivier Gérard

Status

approved