A309280 T(n,k) is (1/k) times the sum of the elements of all subsets of [n] whose sum is divisible by k; triangle T(n,k), n >= 1, 1 <= k <= n*(n+1)/2, read by rows.

1, 6, 1, 1, 24, 6, 4, 1, 1, 1, 80, 20, 9, 4, 4, 2, 2, 1, 1, 1, 240, 60, 30, 14, 12, 7, 5, 3, 3, 3, 2, 2, 1, 1, 1, 672, 168, 84, 42, 29, 20, 15, 10, 9, 7, 5, 5, 4, 4, 4, 3, 2, 2, 1, 1, 1, 1792, 448, 202, 112, 71, 49, 40, 27, 23, 17, 15, 12, 10, 10, 8, 8, 7, 7, 6, 5, 5, 4, 3, 2, 2, 1, 1, 1

Offset

1,2

Comments

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.

The sequence of column k satisfies a linear recurrence with constant coefficients of order 3*A000593(k).

Links

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

Formula

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

Example

The subsets of [4] whose sum is divisible by 3 are: {}, {3}, {1,2}, {2,4}, {1,2,3}, {2,3,4}.  The sum of their elements is 0 + 3 + 3 + 6 + 6 + 9 = 27.  So T(4,3) = 27/3 = 9.
Triangle T(n,k) begins:
    1;
    6,  1,  1;
   24,  6,  4,  1,  1, 1;
   80, 20,  9,  4,  4, 2, 2, 1, 1, 1;
  240, 60, 30, 14, 12, 7, 5, 3, 3, 3, 2, 2, 1, 1, 1;
  ...

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:
T:= (n, k)-> b(n, k, 0)[2]/k:
seq(seq(T(n, k), k=1..n*(n+1)/2), n=1..10);
# second Maple program:
b:= proc(n, s) option remember; `if`(n=0, add(s/d *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);

Mathematica

b[n_, m_, s_] := b[n, m, s] = If[n == 0, {If[s == 0, 1, 0], 0}, b[n-1, m, s] + Function[g, g + {0, g[[1]] n}][b[n-1, m, Mod[s+n, m]]]];
T[n_, k_] := b[n, k, 0][[2]]/k;
Table[T[n, k], {n, 1, 10}, {k, 1, n(n+1)/2}] // Flatten (* Jean-François Alcover, Oct 04 2019, after Alois P. Heinz *)

Crossrefs

Columns k=1..10 give: A001788, A309294, A309295, A309296, A309297, A309298, A309299, A309300, A309301, A309302.

Row sums give A309281.

Row lengths give A000217.

T(n,n) gives A309128.

Rows reversed converge to A000009.

Cf. A000593, A309402.

Sequence in context: A060187 A174527 A156139 * A155863 A173882 A174045

Adjacent sequences: A309277 A309278 A309279 * A309281 A309282 A309283

Keyword

nonn,look,tabf

Author

Alois P. Heinz, Jul 20 2019

Status

approved