A293569 Partitions with designated summands in which no parts are multiples of 3.

1, 1, 3, 4, 9, 12, 21, 29, 48, 64, 99, 132, 195, 257, 366, 480, 666, 864, 1173, 1511, 2016, 2576, 3384, 4296, 5574, 7027, 9015, 11296, 14355, 17880, 22527, 27908, 34896, 43008, 53406, 65508, 80844, 98711, 121128, 147272, 179784, 217704, 264489, 319064

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Formula

Expansion of eta(q^6)^2 * eta(q^9) / (eta(q) * eta(q^2) * eta(q^18)) in powers of q.

a(n) ~ 5^(1/4) * exp(2*Pi*sqrt(5*n/3)/3) / (2 * 3^(7/4)* n^(3/4)). - Vaclav Kotesovec, Oct 13 2017

Example

n = 3        n = 4            n = 5
----------   --------------   ------------------
2'+ 1'       4'               5'
1'+ 1 + 1    2'+ 2            4'+ 1'
1 + 1'+ 1    2 + 2'           2'+ 2 + 1'
1 + 1 + 1'   2'+ 1'+ 1        2 + 2'+ 1'
             2'+ 1 + 1'       2'+ 1'+ 1 + 1
             1'+ 1 + 1 + 1    2'+ 1 + 1'+ 1
             1 + 1'+ 1 + 1    2'+ 1 + 1 + 1'
             1 + 1 + 1'+ 1    1'+ 1 + 1 + 1 + 1
             1 + 1 + 1 + 1'   1 + 1'+ 1 + 1 + 1
                              1 + 1 + 1'+ 1 + 1
                              1 + 1 + 1 + 1'+ 1
                              1 + 1 + 1 + 1 + 1'
----------   --------------   ------------------
a(3) = 4.    a(4) = 9.        a(5) = 12.

Mathematica

nmax = 50; CoefficientList[Series[Product[(1-x^(6*k))^2 / ( (1-x^k)^2 * (1+x^k) * (1+x^(9*k)) ), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2017 *)

Prog

(Ruby)
def partition(n, min, max)
  return [[]] if n == 0
  [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
end
def A(k, n)
  partition(n, 1, n).select{|i| i.all?{|j| j % k > 0}}.map{|a| a.each_with_object(Hash.new(0)){|v, o| o[v] += 1}.values.inject(:*)}.inject(:+)
end
def A293569(n)
  [1] + (1..n).map{|i| A(3, i)}
end
p A293569(40)

Crossrefs

Cf. A077285 (PD(n)), A102186 (PDO(n)), A293629.

Sequence in context: A025613 A356036 A097063 * A304825 A026476 A335604

Adjacent sequences: A293566 A293567 A293568 * A293570 A293571 A293572

Keyword

nonn

Author

Seiichi Manyama, Oct 12 2017

Status

approved