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A293569 Partitions with designated summands in which no parts are multiples of 3. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A230781 A025613 A097063 * A304825 A026476 A002513

Adjacent sequences:  A293566 A293567 A293568 * A293570 A293571 A293572

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Oct 12 2017

STATUS

approved

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Last modified September 22 12:42 EDT 2019. Contains 327307 sequences. (Running on oeis4.)