A293421 The PD_t(n) function (Number of tagged parts over all the partitions of n with designated summands).

1, 3, 6, 13, 24, 45, 77, 132, 213, 346, 537, 834, 1257, 1893, 2778, 4077, 5865, 8421, 11903, 16785, 23364, 32444, 44562, 61041, 82859, 112164, 150639, 201768, 268413, 356100, 469636, 617724, 808236, 1054802, 1370127, 1775286, 2290610, 2948427, 3780717, 4836814

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.: (1/2) * (Product_{k>0} (1 - q^(3*k))^5/((1 - q^k)^3*(1 - q^(6*k))^2) - Product_{k>0} (1 - q^(6*k))/((1 - q^k)*(1 - q^(2*k))*(1 - q^(3*k)))).

a(n) = (1/2) * (A293423(n) - A077285(n)).

a(n) ~ 5^(1/4) * exp(sqrt(10*n)*Pi/3) / (9*2^(5/4)*n^(3/4)). - Vaclav Kotesovec, Oct 15 2017

Example

n = 4
-------------------
4'            -> 1
3'+ 1'        -> 2
2'+ 2         -> 1
2 + 2'        -> 1
2'+ 1'+ 1     -> 2
2'+ 1 + 1'    -> 2
1'+ 1 + 1 + 1 -> 1
1 + 1'+ 1 + 1 -> 1
1 + 1 + 1'+ 1 -> 1
1 + 1 + 1 + 1'-> 1
-------------------
a(4)          = 13.

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).map{|a| a.each_with_object(Hash.new(0)){|v, o| o[v] += 1}.values}.map{|i| i.size * i.inject(:*)}.inject(:+)
end
def A293421(n)
  (1..n).map{|i| A(i)}
end
p A293421(40)

Crossrefs

Cf. A077285 (PD(n)), A293422, A293423.

Sequence in context: A263847 A061567 A293076 * A018081 A001452 A005405

Adjacent sequences: A293418 A293419 A293420 * A293422 A293423 A293424

Keyword

nonn

Author

Seiichi Manyama, Oct 08 2017

Status

approved