A035686 -id:A035686 - cp's OEIS frontend

A035685 Number of partitions of n into parts 8k+2 and 8k+3 with at least one part of each type.

0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 4, 2, 4, 4, 5, 4, 7, 5, 10, 7, 12, 11, 14, 13, 18, 15, 24, 19, 28, 27, 33, 31, 42, 36, 51, 45, 60, 58, 71, 68, 87, 79, 103, 96, 120, 118, 141, 137, 169, 159, 197, 189, 228, 226, 266, 262, 314, 302, 362, 355, 416, 416, 482, 478, 561, 550

Offset: 1

Olivier Gérard

nonn

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

Cf. A035441-A035468, A035618-A035684, A035686-A035699.

nmax = 68; s1 = Range[0, nmax/8]*8 + 2; s2 = Range[0, nmax/8]*8 + 3;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 15 2020 *)
nmax = 68; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 2)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 3)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 15 2020*)

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8 k + 2)))*(-1 + 1/Product_{k>=0} (1 - x^(8 k + 3))). - Robert Price, Aug 15 2020

A035687 Number of partitions of n into parts 8k+2 and 8k+5 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 3, 1, 4, 1, 4, 3, 4, 4, 7, 4, 10, 4, 11, 8, 11, 11, 15, 12, 21, 12, 25, 18, 26, 24, 31, 28, 42, 29, 50, 38, 55, 50, 62, 58, 79, 63, 95, 76, 105, 96, 118, 113, 144, 123, 172, 145, 193, 178, 213, 208, 255, 230, 302, 262, 340, 316

Offset: 1

Olivier Gérard

nonn

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

Cf. A035441-A035468, A035618-A035686, A035688-A035699.

nmax = 70; s1 = Range[0, nmax/8]*8 + 2; s2 = Range[0, nmax/8]*8 + 5;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 15 2020 *)
nmax = 70; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 2)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 5)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 15 2020*)

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8 k + 2)))*(-1 + 1/Product_{k>=0} (1 - x^(8 k + 5))). - Robert Price, Aug 15 2020

A035455 Number of partitions of n into parts 8k+2 or 8k+4.

Original entry on oeis.org

0, 1, 0, 2, 0, 2, 0, 3, 0, 4, 0, 6, 0, 7, 0, 9, 0, 11, 0, 15, 0, 18, 0, 23, 0, 27, 0, 34, 0, 41, 0, 50, 0, 59, 0, 72, 0, 85, 0, 103, 0, 120, 0, 143, 0, 167, 0, 198, 0, 230, 0, 270, 0, 313, 0, 366, 0, 422, 0, 491, 0, 564, 0, 653, 0, 748, 0, 861, 0, 984, 0, 1130, 0, 1287, 0, 1471, 0

Offset: 1

Olivier Gérard

nonn

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

Cf. A035686.

nmax = 100; Rest[CoefficientList[Series[Product[1/((1 - x^(8k+2))*(1 - x^(8k+4))), {k, 0, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 26 2015 *)
nmax = 60; kmax = nmax/8;
s = Flatten[{Range[0, kmax]*8 + 2}~Join~{Range[0, kmax]*8 + 4}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 1, nmax}] (* Robert Price, Aug 03 2020 *)

If n is even, a(n) ~ exp(Pi*sqrt(n/6)) * Gamma(1/4) / (4 * 6^(1/8) * Pi^(3/4) * n^(5/8)). - Vaclav Kotesovec, Aug 26 2015

cp's OEIS Frontend

A035685 Number of partitions of n into parts 8k+2 and 8k+3 with at least one part of each type.

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A035687 Number of partitions of n into parts 8k+2 and 8k+5 with at least one part of each type.

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A035455 Number of partitions of n into parts 8k+2 or 8k+4.

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