cp's OEIS Frontend

A293422 The PDO_t(n) function (Number of tagged parts over all the partitions of n with designated summands in which all parts are odd).

%I A293422 #33 Oct 15 2017 05:47:13
%S A293422 1,2,4,6,10,16,24,36,52,74,104,144,196,264,352,468,614,800,1036,1332,
%T A293422 1704,2168,2744,3456,4331,5408,6724,8328,10278,12640,15496,18936,
%U A293422 23072,28030,33960,41040,49470,59488,71368,85428,102042,121632,144692,171792,203584
%N A293422 The PDO_t(n) function (Number of tagged parts over all the partitions of n with designated summands in which all parts are odd).
%H A293422 Seiichi Manyama, <a href="/A293422/b293422.txt">Table of n, a(n) for n = 1..10000</a>
%H A293422 Bernard L. S. Lin, <a href="https://doi.org/10.1016/j.jnt.2017.08.020">The number of tagged parts over the partitions with designated summands</a>, Journal of Number Theory.
%F A293422 G.f.: q * Product_{k>0} ((1 - q^(2*k))*(1 - q^(3*k))^2*(1 - q^(12*k))^2)/((1 - q^k)^2*(1 - q^(6*k))).
%F A293422 a(n) ~ exp(sqrt(5*n)*Pi/3) / (3 * 2^(3/2) * 5^(1/4) * n^(1/4)). - _Vaclav Kotesovec_, Oct 15 2017
%e A293422 n = 4                 n = 5                     n = 6
%e A293422 -------------------   -----------------------   ---------------------------
%e A293422 3'+ 1'        -> 2    5'                -> 1    5'+ 1'                -> 2
%e A293422 1'+ 1 + 1 + 1 -> 1    3'+ 1'+ 1         -> 2    3'+ 3                 -> 1
%e A293422 1 + 1'+ 1 + 1 -> 1    3'+ 1 + 1'        -> 2    3 + 3'                -> 1
%e A293422 1 + 1 + 1'+ 1 -> 1    1'+ 1 + 1 + 1 + 1 -> 1    3'+ 1'+ 1 + 1         -> 2
%e A293422 1 + 1 + 1 + 1'-> 1    1 + 1'+ 1 + 1 + 1 -> 1    3'+ 1 + 1'+ 1         -> 2
%e A293422                       1 + 1 + 1'+ 1 + 1 -> 1    3'+ 1 + 1 + 1'        -> 2
%e A293422                       1 + 1 + 1 + 1'+ 1 -> 1    1'+ 1 + 1 + 1 + 1 + 1 -> 1
%e A293422                       1 + 1 + 1 + 1 + 1'-> 1    1 + 1'+ 1 + 1 + 1 + 1 -> 1
%e A293422                                                 1 + 1 + 1'+ 1 + 1 + 1 -> 1
%e A293422                                                 1 + 1 + 1 + 1'+ 1 + 1 -> 1
%e A293422                                                 1 + 1 + 1 + 1 + 1'+ 1 -> 1
%e A293422                                                 1 + 1 + 1 + 1 + 1 + 1'-> 1
%e A293422 -------------------   -----------------------   ---------------------------
%e A293422 a(4)          =  6.   a(5)              = 10.   a(6)                  = 16.
%t A293422 nmax = 50; CoefficientList[Series[Product[(1-x^(2*k)) * (1-x^(3*k))^2 * (1-x^(12*k))^2 / ((1-x^k)^2 * (1-x^(6*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 15 2017 *)
%o A293422 (Ruby)
%o A293422 def partition(n, min, max)
%o A293422   return [[]] if n == 0
%o A293422   [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
%o A293422 end
%o A293422 def A(n)
%o A293422   partition(n, 1, n).select{|i| i.all?{|j| j.odd?}}.map{|a| a.each_with_object(Hash.new(0)){|v, o| o[v] += 1}.values}.map{|i| i.size * i.inject(:*)}.inject(:+)
%o A293422 end
%o A293422 def A293422(n)
%o A293422   (1..n).map{|i| A(i)}
%o A293422 end
%o A293422 p A293422(40)
%Y A293422 Cf. A102186 (PDO(n)), A293421.
%K A293422 nonn
%O A293422 1,2
%A A293422 _Seiichi Manyama_, Oct 08 2017