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

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

0,12

Links

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

Formula

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

Example

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

Mathematica

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 *)

Prog

(PARI) 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))

Crossrefs

Cf. A035691.

Sequence in context: A008677 A036497 A211976 * A261679 A147657 A029232

Adjacent sequences: A035457 A035458 A035459 * A035461 A035462 A035463

Keyword

nonn

Author

Olivier Gérard

Status

approved