A293421 - OEIS

%I #43 Oct 10 2025 16:03:36

%S 1,3,6,13,24,45,77,132,213,346,537,834,1257,1893,2778,4077,5865,8421,

%T 11903,16785,23364,32444,44562,61041,82859,112164,150639,201768,

%U 268413,356100,469636,617724,808236,1054802,1370127,1775286,2290610,2948427,3780717,4836814

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

%H Seiichi Manyama, <a href="/A293421/b293421.txt">Table of n, a(n) for n = 1..10000</a>

%H Bernard L. S. Lin, <a href="https://doi.org/10.1016/j.jnt.2017.08.020">The number of tagged parts over the partitions with designated summands</a>, Journal of Number Theory, 2018; 184: 216-234.

%F 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)))).

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

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

%e n = 4

%e -------------------

%e 4'            -> 1

%e 3'+ 1'        -> 2

%e 2'+ 2         -> 1

%e 2 + 2'        -> 1

%e 2'+ 1'+ 1     -> 2

%e 2'+ 1 + 1'    -> 2

%e 1'+ 1 + 1 + 1 -> 1

%e 1 + 1'+ 1 + 1 -> 1

%e 1 + 1 + 1'+ 1 -> 1

%e 1 + 1 + 1 + 1'-> 1

%e -------------------

%e a(4)          = 13.

%p b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i>1, b(n, i-1), 0)+

%p       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)))

%p     end:

%p a:= n-> b(n$2)[2]:

%p seq(a(n), n=1..40);  # _Alois P. Heinz_, Jul 18 2025

%t 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]]}]];

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

%t Table[a[n], {n, 1, 40}] (* _Jean-François Alcover_, Sep 12 2025, after _Alois P. Heinz_ *)

%o (Ruby)

%o def partition(n, min, max)

%o   return [[]] if n == 0

%o   [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}

%o end

%o def A(n)

%o   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(:+)

%o end

%o def A293421(n)

%o   (1..n).map{|i| A(i)}

%o end

%o p A293421(40)

%Y Cf. A077285 (PD(n)), A293422, A293423, A388064.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Oct 08 2017