cp's OEIS Frontend

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.

%I A309402 #26 Jan 27 2021 10:09:38
%S A309402 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,
%T A309402 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,
%U A309402 25,21,19,15,14,12,11,10,9,9,8,8,7,7,6,5,5,4,3,2,2,1,1,1
%N 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.
%C A309402 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.
%H A309402 Alois P. Heinz, <a href="/A309402/b309402.txt">Rows n = 1..50, flattened</a>
%F A309402 Sum_{k=1..n*(n+1)/2} k * T(n,k) = A309281(n).
%F A309402 T(n+1,n*(n+1)/2+1) = A000009(n) for n >= 0.
%e A309402 Triangle T(n,k) begins:
%e A309402    1;
%e A309402    3,  1,  1;
%e A309402    7,  3,  3,  1,  1,  1;
%e A309402   15,  7,  5,  3,  3,  2, 2, 1, 1, 1;
%e A309402   31, 15, 11,  7,  7,  5, 4, 3, 3, 3, 2, 2, 1, 1, 1;
%e A309402   63, 31, 23, 15, 13, 11, 9, 7, 7, 6, 5, 5, 4, 4, 4, 3, 2, 2, 1, 1, 1;
%e A309402   ...
%p A309402 b:= proc(n, s) option remember; `if`(n=0, add(x^d,
%p A309402       d=numtheory[divisors](s)), b(n-1, s)+b(n-1, s+n))
%p A309402     end:
%p A309402 T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n, 0)):
%p A309402 seq(T(n), n=1..10);
%t A309402 b[n_, s_] := b[n, s] = If[n == 0, Sum[x^d,
%t A309402     {d, Divisors[s]}], b[n-1, s] + b[n-1, s+n]];
%t A309402 T[n_] := With[{p = b[n, 0]}, Table[Coefficient[p, x, i],
%t A309402     {i, 1, Exponent[p, x]}]];
%t A309402 Array[T, 10] // Flatten (* _Jean-François Alcover_, Jan 27 2021, after _Alois P. Heinz_ *)
%Y A309402 Column k=1 gives A000225.
%Y A309402 Row sums give A309403.
%Y A309402 Row lengths give A000217.
%Y A309402 T(n,n) gives A082550.
%Y A309402 Rows reversed converge to A000009.
%Y A309402 Cf. A309280, A309281.
%K A309402 nonn,look,tabf
%O A309402 1,2
%A A309402 _Alois P. Heinz_, Jul 28 2019