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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. 5
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 (list; graph; refs; listen; history; text; internal format)
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.

LINKS

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

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.

EXAMPLE

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;

  ...

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 *)

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.

Sequence in context: A248133 A228524 A116407 * A135288 A078026 A126713

Adjacent sequences:  A309399 A309400 A309401 * A309403 A309404 A309405

KEYWORD

nonn,look,tabf

AUTHOR

Alois P. Heinz, Jul 28 2019

STATUS

approved

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Last modified July 25 18:32 EDT 2021. Contains 346291 sequences. (Running on oeis4.)