A035685 -id:A035685 - 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

A035686 Number of partitions of n into parts 8k+2 and 8k+4 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 5, 0, 5, 0, 8, 0, 8, 0, 14, 0, 15, 0, 22, 0, 23, 0, 34, 0, 37, 0, 51, 0, 54, 0, 74, 0, 81, 0, 107, 0, 116, 0, 150, 0, 165, 0, 210, 0, 229, 0, 287, 0, 316, 0, 392, 0, 430, 0, 526, 0, 580, 0, 704, 0, 774, 0, 929, 0, 1024, 0, 1223, 0, 1347, 0

Offset: 1

Olivier Gérard

nonn

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

Cf. A035441-A035468, A035618-A035685, A035687-A035699.

nmax = 77; s1 = Range[0, nmax/8]*8 + 2; 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 15 2020 *)
nmax = 77; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 2)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 4)), {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 + 4))). - Robert Price, Aug 15 2020

A035454 Number of partitions of n into parts 8k+2 or 8k+3.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 8, 8, 10, 11, 13, 13, 16, 16, 19, 20, 23, 25, 29, 30, 35, 37, 41, 44, 50, 53, 60, 64, 72, 76, 85, 90, 100, 107, 118, 126, 140, 148, 163, 174, 190, 203, 223, 237, 260, 277, 301, 321, 349, 371, 403, 430, 466, 496, 537

Offset: 1

Olivier Gérard

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A035685.

nmax:= 100:
g:= mul(1/(1-x^(8*k+2))/(1-x^(8*k+3)),k=0..(nmax-2)/8):
S:= series(g,x,nmax+1):
seq(coeff(S,x,j),j=1..nmax); # Robert Israel, Dec 11 2018

nmax = 100; Rest[CoefficientList[Series[Product[1/((1 - x^(8k+2))*(1 - x^(8k+3))), {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 + 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(1/4) * Gamma(3/8) / (8 * 2^(5/16) * 3^(1/16) * Pi^(11/8) * n^(9/16)). - Vaclav Kotesovec, Aug 26 2015

G.f.: Product_{k>=0} 1/((1-x^(8*k+2))*(1-x^(8*k+3))). - Robert Israel, Dec 11 2018

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.

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A035686 Number of partitions of n into parts 8k+2 and 8k+4 with at least one part of each type.

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A035454 Number of partitions of n into parts 8k+2 or 8k+3.

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