A035697 -id:A035697 - cp's OEIS frontend

A035696 Number of partitions of n into parts 8k+4 and 8k+7 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, 1, 3, 0, 0, 1, 3, 0, 1, 3, 6, 0, 1, 3, 7, 1, 3, 7, 11, 1, 3, 8, 14, 3, 7, 14, 20, 3, 8, 17, 26, 7, 15, 27, 34, 8, 18, 34, 45, 15, 30, 48, 57, 18, 37, 61, 75, 31, 55, 83, 94, 38, 69, 106, 123, 58, 98, 139, 152, 72, 123, 177, 197, 105

Offset: 1

Olivier Gérard

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..5000

Cf. A035441-A035468, A035618-A035695, A035697-A035699.

nmax = 80; s1 = Range[0, nmax/8]*8 + 4; s2 = Range[0, nmax/8]*8 + 7;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 16 2020 *)
nmax = 80; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 4)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 7)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020*)

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8 k + 4)))*(-1 + 1/Product_{k>=0} (1 - x^(8 k + 7))). - Robert Price, Aug 16 2020

A035698 Number of partitions of n into parts 8k+5 and 8k+7 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 1, 3, 3, 1, 2, 1, 3, 5, 3, 6, 5, 3, 5, 3, 8, 9, 7, 11, 8, 8, 10, 9, 15, 16, 15, 19, 15, 16, 20, 19, 28, 28, 27, 33, 27, 32, 36, 36, 48, 48, 49, 55, 48, 57, 64, 65, 82, 79, 83, 92, 83, 100, 106, 112, 134, 129

Offset: 1

Olivier Gérard

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..5000

Cf. A035441-A035468, A035618-A035697, A035699-A035699.

nmax = 81; s1 = Range[0, nmax/8]*8 + 5; s2 = Range[0, nmax/8]*8 + 7;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 16 2020 *)
nmax = 81; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 5)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 7)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020*)

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8 k + 5)))*(-1 + 1/Product_{k>=0} (1 - x^(8 k + 7))). - Robert Price, Aug 16 2020

cp's OEIS Frontend

A035696 Number of partitions of n into parts 8k+4 and 8k+7 with at least one part of each type.

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