A309281 Total sum of the sum of divisors of the element sum over all nonempty subsets of [n].
1, 8, 37, 124, 384, 1088, 2888, 7480, 18764, 45852, 110266, 260935, 609153, 1407089, 3218496, 7298207, 16429096, 36739434, 81668800, 180586647, 397394871, 870673675, 1900033959, 4131237894, 8952390226, 19339847678, 41660216922, 89502201047, 191809609673
Offset: 1
Keywords
Examples
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.
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..600
Programs
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Maple
b:= proc(n, m, s) option remember; `if`(n=0, [`if`(s=0, 1, 0), 0], b(n-1, m, s) +(g-> g+[0, g[1]*n])(b(n-1, m, irem(s+n, m)))) end: a:= n-> add(b(n, k, 0)[2]/k, k=1..n*(n+1)/2): seq(a(n), n=1..22); # second Maple program: b:= proc(n, s) option remember; `if`(n=0, numtheory[sigma](s), b(n-1, s)+b(n-1, s+n)) end: a:= n-> b(n, 0): seq(a(n), n=1..30); -
Mathematica
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]]; a[n_] := b[n, 0]; Array[a, 30] (* Jean-François Alcover, Dec 20 2020, after 2nd Maple program *)
Formula
a(n) = Sum_{k=1..n*(n+1)/2} A309280(n,k).
a(n) = Sum_{k=1..2^n-1} sigma(A096137(n,k)).
a(n) = Sum_{k=1..n*(n+1)/2} sigma(k) * A053632(n,k).
a(n) = Sum_{k=1..n*(n+1)/2} k * A309402(n,k).
a(n) ~ Pi^2 * n^2 * 2^(n-3) / 3. - Vaclav Kotesovec, Aug 05 2019