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

1,2

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.

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

Example

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.

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 *)

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(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)

Crossrefs

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

Sequence in context: A332281 A241903 A261204 * A132002 A098151 A137414

Adjacent sequences: A293419 A293420 A293421 * A293423 A293424 A293425

Keyword

nonn

Author

Seiichi Manyama, Oct 08 2017

Status

approved