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

Author

Seiichi Manyama, Oct 08 2017

Keywords

• nonn

Examples

			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.
		

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

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i>1, b(n, i-1), 0)+
      add((p-> p+[0, p[1]])(b(n-i*j, min(n-i*j, i-1))*j), j=`if`(i=1, n, 1..n/i)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..40);  # Alois P. Heinz, Jul 18 2025

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)

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