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A309281 Total sum of the sum of divisors of the element sum over all nonempty subsets of [n].

%I A309281 #31 Dec 20 2020 12:39:36
%S A309281 1,8,37,124,384,1088,2888,7480,18764,45852,110266,260935,609153,
%T A309281 1407089,3218496,7298207,16429096,36739434,81668800,180586647,
%U A309281 397394871,870673675,1900033959,4131237894,8952390226,19339847678,41660216922,89502201047,191809609673
%N A309281 Total sum of the sum of divisors of the element sum over all nonempty subsets of [n].
%H A309281 Alois P. Heinz, <a href="/A309281/b309281.txt">Table of n, a(n) for n = 1..600</a>
%F A309281 a(n) = Sum_{k=1..n*(n+1)/2} A309280(n,k).
%F A309281 a(n) = Sum_{k=1..2^n-1} sigma(A096137(n,k)).
%F A309281 a(n) = Sum_{k=1..n*(n+1)/2} sigma(k) * A053632(n,k).
%F A309281 a(n) = Sum_{k=1..n*(n+1)/2} k * A309402(n,k).
%F A309281 a(n) ~ Pi^2 * n^2 * 2^(n-3) / 3. - _Vaclav Kotesovec_, Aug 05 2019
%e A309281 The nonempty subsets of [3] are {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}, having element sums 1, 2, 3, 3, 4, 5, 6 with sums of divisors 1, 3, 4, 4, 7, 6, 12, having sum 37.  So a(3) = 37.
%p A309281 b:= proc(n, m, s) option remember; `if`(n=0, [`if`(s=0, 1, 0), 0],
%p A309281       b(n-1, m, s) +(g-> g+[0, g[1]*n])(b(n-1, m, irem(s+n, m))))
%p A309281     end:
%p A309281 a:= n-> add(b(n, k, 0)[2]/k, k=1..n*(n+1)/2):
%p A309281 seq(a(n), n=1..22);
%p A309281 # second Maple program:
%p A309281 b:= proc(n, s) option remember; `if`(n=0,
%p A309281       numtheory[sigma](s), b(n-1, s)+b(n-1, s+n))
%p A309281     end:
%p A309281 a:= n-> b(n, 0):
%p A309281 seq(a(n), n=1..30);
%t A309281 b[n_, s_] := b[n, s] = If[n==0, If[s==0, 0, DivisorSigma[1, s]], b[n-1, s] + b[n-1, s+n]];
%t A309281 a[n_] := b[n, 0];
%t A309281 Array[a, 30] (* _Jean-François Alcover_, Dec 20 2020, after 2nd Maple program *)
%Y A309281 Row sums of A309280.
%Y A309281 Cf. A000203, A000217, A000225, A053632, A096137, A309402, A309403.
%K A309281 nonn
%O A309281 1,2
%A A309281 _Alois P. Heinz_, Jul 20 2019