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