A309281 - OEIS

%I #31 Dec 20 2020 12:39:36

%S 1,8,37,124,384,1088,2888,7480,18764,45852,110266,260935,609153,

%T 1407089,3218496,7298207,16429096,36739434,81668800,180586647,

%U 397394871,870673675,1900033959,4131237894,8952390226,19339847678,41660216922,89502201047,191809609673

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

%H Alois P. Heinz, <a href="/A309281/b309281.txt">Table of n, a(n) for n = 1..600</a>

%F a(n) = Sum_{k=1..n*(n+1)/2} A309280(n,k).

%F a(n) = Sum_{k=1..2^n-1} sigma(A096137(n,k)).

%F a(n) = Sum_{k=1..n*(n+1)/2} sigma(k) * A053632(n,k).

%F a(n) = Sum_{k=1..n*(n+1)/2} k * A309402(n,k).

%F a(n) ~ Pi^2 * n^2 * 2^(n-3) / 3. - _Vaclav Kotesovec_, Aug 05 2019

%e 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 b:= proc(n, m, s) option remember; `if`(n=0, [`if`(s=0, 1, 0), 0],

%p       b(n-1, m, s) +(g-> g+[0, g[1]*n])(b(n-1, m, irem(s+n, m))))

%p     end:

%p a:= n-> add(b(n, k, 0)[2]/k, k=1..n*(n+1)/2):

%p seq(a(n), n=1..22);

%p # second Maple program:

%p b:= proc(n, s) option remember; `if`(n=0,

%p       numtheory[sigma](s), b(n-1, s)+b(n-1, s+n))

%p     end:

%p a:= n-> b(n, 0):

%p seq(a(n), n=1..30);

%t 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 a[n_] := b[n, 0];

%t Array[a, 30] (* _Jean-François Alcover_, Dec 20 2020, after 2nd Maple program *)

%Y Row sums of A309280.

%Y Cf. A000203, A000217, A000225, A053632, A096137, A309402, A309403.

%K nonn

%O 1,2

%A _Alois P. Heinz_, Jul 20 2019