cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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.

Original entry on oeis.org

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

Views

Author

Alois P. Heinz, Jul 20 2019

Keywords

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).

Examples

			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;
  ...

Links

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.

Programs

  • 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 *)

Formula

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

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.

Original entry on oeis.org

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

Views

Author

Alois P. Heinz, Jul 28 2019

Keywords

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.

Examples

			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;
  ...

Links

Crossrefs

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.

Programs

  • Maple
    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);
  • Mathematica
    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 *)

Formula

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.

A309403 Total sum of the number of divisors of the element sum over all nonempty subsets of [n].

Original entry on oeis.org

1, 5, 16, 40, 96, 217, 469, 1011, 2147, 4497, 9389, 19489, 40256, 82948, 170413, 349158, 714153, 1458199, 2972683, 6052561, 12308971, 25006177, 50755272, 102933086, 208594116, 422432018, 854956112, 1729360940, 3496259940, 7065053883, 14270420877, 28812580857
Offset: 1

Views

Author

Alois P. Heinz, Jul 28 2019

Keywords

Links

Crossrefs

Row sums of A309402.
Cf. A000005, A309281.

Programs

  • Maple
    b:= proc(n, s) option remember; `if`(n=0,
      numtheory[tau](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[0, s]], b[n-1, s] + b[n-1, s+n]];
a[n_] := b[n, 0];
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 24 2022, after Alois P. Heinz *)
Showing 1-3 of 3 results.

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