A309128 -id:A309128 - cp's OEIS frontend

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

Alois P. Heinz, Jul 20 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.

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

			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;
  ...

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

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.

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);

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 *)

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

A309122 Sum of the sizes of all subsets of [n] whose sum is divisible by n.

Original entry on oeis.org

1, 1, 6, 6, 20, 34, 70, 124, 270, 516, 1034, 2060, 4108, 8198, 16440, 32760, 65552, 131142, 262162, 524312, 1048740, 2097162, 4194326, 8388856, 16777300, 33554444, 67109418, 134217764, 268435484, 536872072, 1073741854, 2147483632, 4294969404, 8589934608

Offset: 1

Alois P. Heinz, Jul 13 2019

nonn

The bivariate g.f. of array T(n,k) = A267632(n,k) is Sum_{n, k >= 1} T(n,k) * x^n * y^k = -x/(1 - x) - Sum_{s >= 1} (phi(s)/s) * log(1 - x^s + (-x*y)^s). Differentiating w.r.t. y and setting y = 1, we get the g.f. of a(n) = k * Sum_{1 <= k <= n} T(n,k) (see below). - Petros Hadjicostas, Jul 13 2019

			a(5) = 20 = 0 + 1 + 2 + 2 + 3 + 3 + 4 + 5 = |{}| + |{5}| + |{1,4}| + |{2,3}| + |{1,4,5}| + |{2,3,5}| + |{1,2,3,4}| + |{1,2,3,4,5}|.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000010, A053636, A267632, A309128.

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]])(b(n-1, m, irem(s+n, m))))
    end:
a:= proc(n) option remember; forget(b); b(n$2, 0)[2] end:
seq(a(n), n=1..40);

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]]}][b[n-1, m, Mod[s+n, m]]]];
a[n_] := b[n, n, 0][[2]];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Mar 19 2022, after Alois P. Heinz *)

a(n) = Sum_{k=1..n} k * A267632(n,k).
From Petros Hadjicostas, Jul 13 2019: (Start)
G.f.: Sum_{s >= 1} phi(s) * (-x)^(s-1)/(1 - x^s + (-x)^s) = -Sum_{m >= 1} phi(2*m) * x^(2*m-1) + Sum_{m >= 0} phi(2*m+1) * x^(2*m)/(1 - 2*x^(2*m+1)).
a(2*m + 1) = A053636(2*m + 1)/2 = (1/2) * Sum_{d|2*m+1} phi(d) * 2^((2*m+1)/d) for m >= 0.
a(2*m) = -phi(2*m) +  A053636(2*m)/2 for m >= 1.
(End)

cp's OEIS Frontend

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.

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