cp's OEIS Frontend

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

%I A293421 #35 Jul 18 2025 20:24:16
%S A293421 1,3,6,13,24,45,77,132,213,346,537,834,1257,1893,2778,4077,5865,8421,
%T A293421 11903,16785,23364,32444,44562,61041,82859,112164,150639,201768,
%U A293421 268413,356100,469636,617724,808236,1054802,1370127,1775286,2290610,2948427,3780717,4836814
%N A293421 The PD_t(n) function (Number of tagged parts over all the partitions of n with designated summands).
%H A293421 Seiichi Manyama, <a href="/A293421/b293421.txt">Table of n, a(n) for n = 1..10000</a>
%H A293421 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.
%F A293421 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 A293421 a(n) = (1/2) * (A293423(n) - A077285(n)).
%F A293421 a(n) ~ 5^(1/4) * exp(sqrt(10*n)*Pi/3) / (9*2^(5/4)*n^(3/4)). - _Vaclav Kotesovec_, Oct 15 2017
%e A293421 n = 4
%e A293421 -------------------
%e A293421 4'            -> 1
%e A293421 3'+ 1'        -> 2
%e A293421 2'+ 2         -> 1
%e A293421 2 + 2'        -> 1
%e A293421 2'+ 1'+ 1     -> 2
%e A293421 2'+ 1 + 1'    -> 2
%e A293421 1'+ 1 + 1 + 1 -> 1
%e A293421 1 + 1'+ 1 + 1 -> 1
%e A293421 1 + 1 + 1'+ 1 -> 1
%e A293421 1 + 1 + 1 + 1'-> 1
%e A293421 -------------------
%e A293421 a(4)          = 13.
%p A293421 b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i>1, b(n, i-1), 0)+
%p A293421       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 A293421     end:
%p A293421 a:= n-> b(n$2)[2]:
%p A293421 seq(a(n), n=1..40);  # _Alois P. Heinz_, Jul 18 2025
%o A293421 (Ruby)
%o A293421 def partition(n, min, max)
%o A293421   return [[]] if n == 0
%o A293421   [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
%o A293421 end
%o A293421 def A(n)
%o A293421   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 A293421 end
%o A293421 def A293421(n)
%o A293421   (1..n).map{|i| A(i)}
%o A293421 end
%o A293421 p A293421(40)
%Y A293421 Cf. A077285 (PD(n)), A293422, A293423.
%K A293421 nonn
%O A293421 1,2
%A A293421 _Seiichi Manyama_, Oct 08 2017