A035691 -id:A035691 - cp's OEIS frontend

A035690 Number of partitions of n into parts 8k+3 and 8k+4 with at least one part of each type.

0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 3, 1, 1, 3, 4, 1, 3, 4, 7, 3, 4, 8, 10, 4, 8, 11, 15, 8, 11, 18, 21, 11, 19, 24, 30, 19, 25, 37, 42, 25, 40, 50, 56, 41, 53, 70, 79, 54, 77, 95, 103, 80, 103, 129, 141, 106, 144, 172, 183, 151, 189, 228, 246, 197, 257, 301, 314

Offset: 1

Olivier Gérard

nonn

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

Cf. A035441-A035468, A035618-A035689, A035691-A035699.

nmax = 71; s1 = Range[0, nmax/8]*8 + 3; 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 = 71; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 3)), {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 + 3)))*(-1 + 1/Product_{k>=0} (1 - x^(8 k + 4))). - Robert Price, Aug 15 2020

A035692 Number of partitions of n into parts 8k+3 and 8k+6 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 2, 2, 0, 2, 3, 0, 4, 3, 3, 4, 4, 4, 6, 4, 8, 6, 9, 9, 8, 11, 13, 8, 19, 14, 15, 21, 18, 19, 30, 19, 32, 32, 29, 38, 41, 36, 53, 43, 56, 59, 59, 67, 75, 70, 93, 81, 102, 105, 105, 122, 133, 123, 165, 145, 170, 189, 183, 203, 237

Offset: 1

Olivier Gérard

nonn

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

Cf. A035441-A035468, A035618-A035691, A035693-A035699.

nmax = 75; s1 = Range[0, nmax/8]*8 + 3; s2 = Range[0, nmax/8]*8 + 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 = 75; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 3)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 6)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 15 2020*)

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8 k + 3)))*(-1 + 1/Product_{k>=0} (1 - x^(8 k + 6))). - Robert Price, Aug 15 2020

A035460 Number of partitions of n into parts 8k+3 or 8k+5.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 2, 1, 2, 2, 2, 3, 2, 3, 4, 3, 5, 5, 4, 7, 6, 7, 9, 7, 10, 11, 10, 14, 13, 14, 18, 16, 20, 22, 21, 27, 26, 29, 34, 32, 39, 41, 41, 51, 49, 54, 63, 60, 71, 76, 76, 90, 91, 98, 111, 110, 125, 133, 137, 157, 159, 172, 191, 192, 216, 229, 235, 266, 272

Offset: 0

Olivier Gérard

nonn

			G.f. = 1 + x^3 + x^5 + x^6 + x^8 + x^9 + x^10 + 2*x^11 + x^12 + 2*x^13 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A035691.

a[ n_] := SeriesCoefficient[ 1 / QPochhammer[ x^3, x^8] / QPochhammer[ x^5, x^8], {x, 0, n}]; (* Michael Somos, Jun 03 2014 *)
nmax = 100; CoefficientList[Series[Product[1/((1 - x^(8k+3))*(1 - x^(8k+5))), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 26 2015 *)
nmax = 60; kmax = nmax/8;
s = Flatten[{Range[0, kmax]*8 + 3}~Join~{Range[0, kmax]*8 + 5}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 04 2020 *)

a(n)=if(n<0,0,polcoeff(1/prod(k=1, n, 1-(k%8==3||k%8==5)*x^k, 1+x*O(x^n)), n))

Expansion of f(-x^8) / f(-x^3, -x^5) in powers of x where f() is a Ramanujan theta function. - Michael Somos, Jun 03 2014
Euler transform of period 8 sequence [ 0, 0, 1, 0, 1, 0, 0, 0, ...]. - Michael Somos, Jun 03 2014
a(n) ~ (3-2*sqrt(2))^(1/4) * exp(Pi*sqrt(n/6)) / (4 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 26 2015

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A035690 Number of partitions of n into parts 8k+3 and 8k+4 with at least one part of each type.

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A035692 Number of partitions of n into parts 8k+3 and 8k+6 with at least one part of each type.

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A035460 Number of partitions of n into parts 8k+3 or 8k+5.

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