A035683 -id:A035683 - cp's OEIS frontend

A035684 Number of partitions of n into parts 8k+1 and 8k+7 with at least one part of each type.

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 4, 4, 4, 4, 4, 4, 5, 7, 10, 11, 11, 11, 11, 12, 14, 18, 23, 25, 26, 26, 27, 29, 33, 40, 47, 52, 54, 56, 58, 62, 70, 81, 93, 101, 107, 111, 116, 124, 137, 155, 172, 188, 199, 208, 218, 233, 255, 282, 311, 336, 357, 374, 393

Offset: 1

Olivier Gérard

nonn

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

Cf. A035441-A035468, A035618-A035683, A035685-A035699.

nmax = 68; s1 = Range[0, nmax/8]*8 + 1; 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 = 68; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 1)), {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 + 1)))*(-1 + 1/Product_{k>=0} (1 - x^(8*k + 7))). - Robert Price, Aug 15 2020

A035682 Number of partitions of n into parts 8k+1 and 8k+5 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 4, 4, 5, 5, 5, 7, 7, 8, 11, 12, 14, 14, 15, 19, 20, 22, 26, 29, 34, 35, 37, 43, 46, 51, 57, 63, 72, 76, 81, 91, 98, 107, 117, 128, 144, 153, 163, 179, 192, 210, 226, 245, 272, 290, 310, 336, 360, 391, 418, 450, 494, 527, 564, 605

Offset: 1

Olivier Gérard

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A035441-A035468, A035618-A035681, A035683-A035699.

nmax = 67; s1 = Range[0, nmax/8]*8 + 1; 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 = 67; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 1)), {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 + 1)))*(-1 + 1/Product_{k>=0} (1 - x^(8*k + 5))). - Robert Price, Aug 15 2020

A035452 Number of partitions of n into parts 8k+1 or 8k+6.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 5, 6, 6, 7, 9, 9, 10, 11, 12, 14, 16, 17, 19, 21, 23, 25, 28, 31, 34, 37, 41, 44, 49, 53, 57, 63, 69, 74, 82, 88, 95, 104, 112, 121, 133, 142, 154, 167, 179, 193, 209, 224, 243, 261, 280, 301, 324, 347, 373, 400, 430, 460, 494, 528

Offset: 1

Olivier Gérard

nonn

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

Cf. A035683.

nmax = 100; Rest[CoefficientList[Series[Product[1/((1 - x^(8k+1))*(1 - x^(8k+6))), {k, 0, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 26 2015 *)
nmax = 60; kmax = nmax/8;
s = Flatten[{Range[0, kmax]*8 + 1}~Join~{Range[0, kmax]*8 + 6}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 1, nmax}] (* Robert Price, Aug 03 2020 *)

a(n) ~ exp(Pi*sqrt(n/6)) * Gamma(3/4) * Gamma(1/8) / (4 * 2^(15/16) * 3^(3/16) * Pi^(9/8) * n^(11/16)). - Vaclav Kotesovec, Aug 26 2015

cp's OEIS Frontend

A035684 Number of partitions of n into parts 8k+1 and 8k+7 with at least one part of each type.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A035682 Number of partitions of n into parts 8k+1 and 8k+5 with at least one part of each type.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A035452 Number of partitions of n into parts 8k+1 or 8k+6.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula