A293421 - OEIS

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

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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, 2018; 184: 216-234.

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.

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

MATHEMATICA

b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i > 1, b[n, i-1], 0] + Sum[Function[p, p + {0, p[[1]]}][b[n-i*j, Min[n-i*j, i-1]]*j], {j, If[i == 1, {n}, Range[n/i]]}]];

a[n_] := b[n, n][[2]];

Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Sep 12 2025, after Alois P. Heinz *)

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)