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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

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%I A293569 #26 Oct 14 2017 03:34:36
%S A293569 1,1,3,4,9,12,21,29,48,64,99,132,195,257,366,480,666,864,1173,1511,
%T A293569 2016,2576,3384,4296,5574,7027,9015,11296,14355,17880,22527,27908,
%U A293569 34896,43008,53406,65508,80844,98711,121128,147272,179784,217704,264489,319064
%N A293569 Partitions with designated summands in which no parts are multiples of 3.
%H A293569 Seiichi Manyama, <a href="/A293569/b293569.txt">Table of n, a(n) for n = 0..10000</a>
%F A293569 Expansion of eta(q^6)^2 * eta(q^9) / (eta(q) * eta(q^2) * eta(q^18)) in powers of q.
%F A293569 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
%e A293569 n = 3        n = 4            n = 5
%e A293569 ----------   --------------   ------------------
%e A293569 2'+ 1'       4'               5'
%e A293569 1'+ 1 + 1    2'+ 2            4'+ 1'
%e A293569 1 + 1'+ 1    2 + 2'           2'+ 2 + 1'
%e A293569 1 + 1 + 1'   2'+ 1'+ 1        2 + 2'+ 1'
%e A293569              2'+ 1 + 1'       2'+ 1'+ 1 + 1
%e A293569              1'+ 1 + 1 + 1    2'+ 1 + 1'+ 1
%e A293569              1 + 1'+ 1 + 1    2'+ 1 + 1 + 1'
%e A293569              1 + 1 + 1'+ 1    1'+ 1 + 1 + 1 + 1
%e A293569              1 + 1 + 1 + 1'   1 + 1'+ 1 + 1 + 1
%e A293569                               1 + 1 + 1'+ 1 + 1
%e A293569                               1 + 1 + 1 + 1'+ 1
%e A293569                               1 + 1 + 1 + 1 + 1'
%e A293569 ----------   --------------   ------------------
%e A293569 a(3) = 4.    a(4) = 9.        a(5) = 12.
%t A293569 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 *)
%o A293569 (Ruby)
%o A293569 def partition(n, min, max)
%o A293569   return [[]] if n == 0
%o A293569   [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
%o A293569 end
%o A293569 def A(k, n)
%o A293569   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(:+)
%o A293569 end
%o A293569 def A293569(n)
%o A293569   [1] + (1..n).map{|i| A(3, i)}
%o A293569 end
%o A293569 p A293569(40)
%Y A293569 Cf. A077285 (PD(n)), A102186 (PDO(n)), A293629.
%K A293569 nonn
%O A293569 0,3
%A A293569 _Seiichi Manyama_, Oct 12 2017