A309403 -id:A309403 - cp's OEIS frontend

A309402 Number T(n,k) of nonempty subsets of [n] whose element sum is divisible by k; triangle T(n,k), n >= 1, 1 <= k <= n*(n+1)/2, read by rows.

1, 3, 1, 1, 7, 3, 3, 1, 1, 1, 15, 7, 5, 3, 3, 2, 2, 1, 1, 1, 31, 15, 11, 7, 7, 5, 4, 3, 3, 3, 2, 2, 1, 1, 1, 63, 31, 23, 15, 13, 11, 9, 7, 7, 6, 5, 5, 4, 4, 4, 3, 2, 2, 1, 1, 1, 127, 63, 43, 31, 25, 21, 19, 15, 14, 12, 11, 10, 9, 9, 8, 8, 7, 7, 6, 5, 5, 4, 3, 2, 2, 1, 1, 1

Offset: 1

Alois P. Heinz, Jul 28 2019

T(n,k) is defined for all n >= 0, k >= 1. The triangle contains only the positive terms. T(n,k) = 0 if k > n*(n+1)/2.

			Triangle T(n,k) begins:
   1;
   3,  1,  1;
   7,  3,  3,  1,  1,  1;
  15,  7,  5,  3,  3,  2, 2, 1, 1, 1;
  31, 15, 11,  7,  7,  5, 4, 3, 3, 3, 2, 2, 1, 1, 1;
  63, 31, 23, 15, 13, 11, 9, 7, 7, 6, 5, 5, 4, 4, 4, 3, 2, 2, 1, 1, 1;
  ...

Alois P. Heinz, Rows n = 1..50, flattened

Column k=1 gives A000225.
Row sums give A309403.
Row lengths give A000217.
T(n,n) gives A082550.
Rows reversed converge to A000009.
Cf. A309280, A309281.

b:= proc(n, s) option remember; `if`(n=0, add(x^d,
      d=numtheory[divisors](s)), b(n-1, s)+b(n-1, s+n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n, 0)):
seq(T(n), n=1..10);

b[n_, s_] := b[n, s] = If[n == 0, Sum[x^d,
    {d, Divisors[s]}], b[n-1, s] + b[n-1, s+n]];
T[n_] := With[{p = b[n, 0]}, Table[Coefficient[p, x, i],
    {i, 1, Exponent[p, x]}]];
Array[T, 10] // Flatten (* Jean-François Alcover, Jan 27 2021, after Alois P. Heinz *)

Sum_{k=1..n*(n+1)/2} k * T(n,k) = A309281(n).

T(n+1,n*(n+1)/2+1) = A000009(n) for n >= 0.

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

Original entry on oeis.org

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

Alois P. Heinz, Jul 20 2019

nonn

			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.

Alois P. Heinz, Table of n, a(n) for n = 1..600

Row sums of A309280.

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

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);

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 *)

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

cp's OEIS Frontend

A309402 Number T(n,k) of nonempty subsets of [n] whose element sum is divisible by k; triangle T(n,k), n >= 1, 1 <= k <= n*(n+1)/2, read by rows.

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