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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
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
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.
Sequence in context: A346906 A228524 A116407 * A135288 A078026 A350635
KEYWORD
nonn,look,tabf
AUTHOR
Alois P. Heinz, Jul 28 2019
STATUS
approved