A035672 -id:A035672 - cp's OEIS frontend

A035441 Number of partitions of n into parts 8k or 8k+1.

1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 3, 3, 3, 3, 3, 5, 7, 8, 8, 8, 8, 8, 8, 11, 15, 17, 18, 18, 18, 18, 18, 23, 30, 35, 37, 38, 38, 38, 38, 45, 57, 66, 71, 73, 74, 74, 74, 85, 104, 121, 131, 136, 138, 139, 139, 154, 184, 212, 231, 241, 246, 248, 249, 271, 316, 363, 396, 416

Offset: 1

Olivier Gérard

nonn

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

Cf. A035672.

nmax = 100; Rest[CoefficientList[Series[Product[1/((1 - x^(8k+8))*(1 - x^(8k+1))), {k, 0, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 26 2015 *)
nmax = 68; kmax = nmax/8;
s = Flatten[{Range[0, kmax]*8}~Join~{Range[0, kmax]*8 + 1}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 1, nmax}] (* Robert Price, Aug 02 2020 *)

a(n) ~ exp(Pi*sqrt(n/6)) * Gamma(9/8) * 2^(7/16) / (3^(5/16) * Pi^(7/8) * n^(13/16)). - Vaclav Kotesovec, Aug 26 2015

A035671 Number of partitions of n into parts 7k+5 and 7k+6 with at least one part of each type.

Original entry on oeis.org

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

Olivier Gérard

nonn

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

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

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 *)

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

A035673 Number of partitions of n into parts 8k and 8k+2 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 4, 0, 4, 0, 4, 0, 4, 0, 10, 0, 11, 0, 11, 0, 11, 0, 22, 0, 25, 0, 26, 0, 26, 0, 44, 0, 51, 0, 54, 0, 55, 0, 84, 0, 98, 0, 105, 0, 108, 0, 153, 0, 178, 0, 193, 0, 200, 0, 269, 0, 313, 0, 341, 0, 356, 0, 459, 0, 531, 0, 582, 0, 611, 0

Offset: 1

Olivier Gérard

nonn

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

Cf. A035441-A035468, A035618-A035672, A035674-A035699.

Bisections give: A035621 (even part), A000004 (odd part).

nmax = 81; s1 = Range[1, nmax/8]*8; s2 = Range[0, nmax/8]*8 + 2;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 12 2020 *)
nmax = 81; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 2)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 12 2020 *)

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8*k + 2)))*(-1 + 1/Product_{k>=1} (1 - x^(8*k))). - Robert Price, Aug 12 2020

cp's OEIS Frontend

A035441 Number of partitions of n into parts 8k or 8k+1.

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A035671 Number of partitions of n into parts 7k+5 and 7k+6 with at least one part of each type.

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A035673 Number of partitions of n into parts 8k and 8k+2 with at least one part of each type.

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