A035674 -id:A035674 - cp's OEIS frontend

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

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

A035675 Number of partitions of n into parts 8k and 8k+4 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, 1, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 10, 0, 0, 0, 11, 0, 0, 0, 22, 0, 0, 0, 25, 0, 0, 0, 44, 0, 0, 0, 51, 0, 0, 0, 83, 0, 0, 0, 98, 0, 0, 0, 149, 0, 0, 0, 177, 0, 0, 0, 259, 0, 0, 0, 309, 0, 0, 0, 436, 0, 0, 0, 521, 0, 0, 0, 716, 0, 0, 0, 857, 0, 0

Offset: 1

Olivier Gérard

nonn

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

Cf. A035441-A035468, A035618-A035674, A035676-A035699.

nmax = 90; s1 = Range[1, nmax/8]*8; s2 = Range[0, nmax/8]*8 + 4;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 13 2020 *)
nmax = 90; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 4)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 13 2020 *)

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8*k + 4)))*(-1 + 1/Product_{k>=1} (1 - x^(8*k))). - Robert Price, Aug 13 2020

A035443 Number of partitions of n into parts 8k or 8k+3.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 2, 1, 2, 2, 1, 4, 2, 1, 5, 2, 4, 5, 2, 8, 5, 2, 10, 5, 7, 11, 5, 14, 11, 5, 19, 11, 12, 21, 11, 24, 22, 11, 33, 22, 22, 38, 22, 41, 40, 22, 58, 41, 37, 68, 41, 67, 73, 41, 95, 75, 63, 114, 76, 108, 124, 76, 155, 129, 106, 188, 131, 173

Offset: 1

Olivier Gérard

nonn

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

Cf. A035674.

nmax = 100; Rest[CoefficientList[Series[Product[1/((1 - x^(8k+8))*(1 - x^(8k+3))), {k, 0, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 26 2015 *)
nmax = 50; kmax = nmax/8;
s = Flatten[{Range[0, kmax]*8}~Join~{Range[0, kmax]*8 + 3}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 1, nmax}] (* Robert Price, Aug 03 2020 *)

a(n) ~ exp(Pi*sqrt(n/6)) * Gamma(3/8) / (4 * 2^(3/16) * 3^(7/16) * Pi^(5/8) * n^(15/16)). - Vaclav Kotesovec, Aug 26 2015

cp's OEIS Frontend

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

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A035675 Number of partitions of n into parts 8k and 8k+4 with at least one part of each type.

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A035443 Number of partitions of n into parts 8k or 8k+3.

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