A035694 -id:A035694 - cp's OEIS frontend

A035695 Number of partitions of n into parts 8k+4 and 8k+6 with at least one part of each type.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 3, 0, 1, 0, 4, 0, 3, 0, 7, 0, 4, 0, 10, 0, 8, 0, 15, 0, 11, 0, 21, 0, 18, 0, 30, 0, 24, 0, 42, 0, 37, 0, 56, 0, 50, 0, 78, 0, 70, 0, 102, 0, 95, 0, 137, 0, 129, 0, 179, 0, 171, 0, 236, 0, 227, 0, 303, 0, 297, 0, 395, 0, 386, 0, 502, 0

Offset: 1

Olivier Gérard

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Bisections give A035626 (even part), A000004 (odd part).

Cf. A035441-A035468, A035618-A035694, A035696-A035699.

nmax = 83; s1 = Range[0, nmax/8]*8 + 4; s2 = Range[0, nmax/8]*8 + 6;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 16 2020 *)
nmax = 83; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 4)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 6)), {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 + 6))). - Robert Price, Aug 16 2020

A035693 Number of partitions of n into parts 8k+3 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, 1, 0, 0, 1, 0, 0, 1, 1, 2, 1, 1, 2, 1, 1, 3, 3, 4, 3, 3, 5, 3, 4, 7, 7, 8, 7, 8, 10, 8, 10, 15, 14, 15, 16, 16, 20, 18, 21, 29, 27, 28, 32, 33, 36, 37, 42, 51, 50, 52, 59, 60, 66, 69, 77, 90, 88, 93, 105, 107, 115, 124, 135, 152, 153, 160, 180, 183, 195

Offset: 1

Olivier Gérard

nonn

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

Cf. A035441-A035468, A035618-A035693, A035694-A035699.

nmax = 77; s1 = Range[0, nmax/8]*8 + 3; 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 15 2020 *)
nmax = 77; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 3)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 7)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 15 2020*)

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8 k + 3)))*(-1 + 1/Product_{k>=0} (1 - x^(8 k + 7))). - Robert Price, Aug 15 2020

A035463 Number of partitions of n into parts 8k+4 or 8k+5.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 2, 2, 1, 1, 2, 3, 2, 1, 4, 4, 3, 2, 5, 7, 5, 3, 7, 9, 8, 5, 9, 13, 11, 8, 13, 17, 16, 12, 18, 24, 22, 17, 24, 32, 31, 24, 32, 43, 42, 34, 43, 56, 57, 47, 57, 74, 75, 64, 76, 96, 100, 86, 99, 126, 130, 115, 129, 161, 171, 151, 168, 207, 219, 200

Offset: 1

Olivier Gérard

nonn

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

Cf. A035694.

nmax = 100; Rest[CoefficientList[Series[Product[1/((1 - x^(8k+4))*(1 - x^(8k+5))), {k, 0, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 27 2015 *)
nmax = 60; kmax = nmax/8;
s = Flatten[{Range[0, kmax]*8 + 4}~Join~{Range[0, kmax]*8 + 5}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 1, nmax}] (* Robert Price, Aug 04 2020 *)

a(n) ~ exp(Pi*sqrt(n/6)) * Gamma(5/8) / (4 * 2^(9/16) * 3^(5/16) * Pi^(3/8) * n^(13/16)). - Vaclav Kotesovec, Aug 27 2015

cp's OEIS Frontend

A035695 Number of partitions of n into parts 8k+4 and 8k+6 with at least one part of each type.

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A035693 Number of partitions of n into parts 8k+3 and 8k+7 with at least one part of each type.

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A035463 Number of partitions of n into parts 8k+4 or 8k+5.

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