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).

1, 2, 4, 6, 10, 16, 24, 36, 52, 74, 104, 144, 196, 264, 352, 468, 614, 800, 1036, 1332, 1704, 2168, 2744, 3456, 4331, 5408, 6724, 8328, 10278, 12640, 15496, 18936, 23072, 28030, 33960, 41040, 49470, 59488, 71368, 85428, 102042, 121632, 144692, 171792, 203584

Offset: 1

Author

Seiichi Manyama, Oct 08 2017

Keywords

• nonn

Examples

			n = 4                 n = 5                     n = 6
-------------------   -----------------------   ---------------------------
3'+ 1'        -> 2    5'                -> 1    5'+ 1'                -> 2
1'+ 1 + 1 + 1 -> 1    3'+ 1'+ 1         -> 2    3'+ 3                 -> 1
1 + 1'+ 1 + 1 -> 1    3'+ 1 + 1'        -> 2    3 + 3'                -> 1
1 + 1 + 1'+ 1 -> 1    1'+ 1 + 1 + 1 + 1 -> 1    3'+ 1'+ 1 + 1         -> 2
1 + 1 + 1 + 1'-> 1    1 + 1'+ 1 + 1 + 1 -> 1    3'+ 1 + 1'+ 1         -> 2
                      1 + 1 + 1'+ 1 + 1 -> 1    3'+ 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 + 1'+ 1 + 1 -> 1
                                                1 + 1 + 1 + 1 + 1'+ 1 -> 1
                                                1 + 1 + 1 + 1 + 1 + 1'-> 1
-------------------   -----------------------   ---------------------------
a(4)          =  6.   a(5)              = 10.   a(6)                  = 16.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• Bernard L. S. Lin, The number of tagged parts over the partitions with designated summands, Journal of Number Theory.

Crossrefs

Cf. A102186 (PDO(n)), A293421.

Programs

Mathematica

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 *)

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(n)
  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(:+)
end
def A293422(n)
  (1..n).map{|i| A(i)}
end
p A293422(40)

Formula

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))).

a(n) ~ exp(sqrt(5*n)*Pi/3) / (3 * 2^(3/2) * 5^(1/4) * n^(1/4)). - Vaclav Kotesovec, Oct 15 2017