cp's OEIS Frontend

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.

%I A309280 #32 Oct 04 2019 13:10:15
%S A309280 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,
%T A309280 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,
%U A309280 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
%N 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.
%C A309280 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.
%C A309280 The sequence of column k satisfies a linear recurrence with constant coefficients of order 3*A000593(k).
%H A309280 Alois P. Heinz, <a href="/A309280/b309280.txt">Rows n = 1..50, flattened</a>
%F A309280 T(n+1,n*(n+1)/2+1) = A000009(n) for n >= 0.
%e A309280 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.
%e A309280 Triangle T(n,k) begins:
%e A309280     1;
%e A309280     6,  1,  1;
%e A309280    24,  6,  4,  1,  1, 1;
%e A309280    80, 20,  9,  4,  4, 2, 2, 1, 1, 1;
%e A309280   240, 60, 30, 14, 12, 7, 5, 3, 3, 3, 2, 2, 1, 1, 1;
%e A309280   ...
%p A309280 b:= proc(n, m, s) option remember; `if`(n=0, [`if`(s=0, 1, 0), 0],
%p A309280       b(n-1, m, s) +(g-> g+[0, g[1]*n])(b(n-1, m, irem(s+n, m))))
%p A309280     end:
%p A309280 T:= (n, k)-> b(n, k, 0)[2]/k:
%p A309280 seq(seq(T(n, k), k=1..n*(n+1)/2), n=1..10);
%p A309280 # second Maple program:
%p A309280 b:= proc(n, s) option remember; `if`(n=0, add(s/d *x^d,
%p A309280       d=numtheory[divisors](s)), b(n-1, s)+b(n-1, s+n))
%p A309280     end:
%p A309280 T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n, 0)):
%p A309280 seq(T(n), n=1..10);
%t A309280 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 A309280 T[n_, k_] := b[n, k, 0][[2]]/k;
%t A309280 Table[T[n, k], {n, 1, 10}, {k, 1, n(n+1)/2}] // Flatten (* _Jean-François Alcover_, Oct 04 2019, after _Alois P. Heinz_ *)
%Y A309280 Columns k=1..10 give: A001788, A309294, A309295, A309296, A309297, A309298, A309299, A309300, A309301, A309302.
%Y A309280 Row sums give A309281.
%Y A309280 Row lengths give A000217.
%Y A309280 T(n,n) gives A309128.
%Y A309280 Rows reversed converge to A000009.
%Y A309280 Cf. A000593, A309402.
%K A309280 nonn,look,tabf
%O A309280 1,2
%A A309280 _Alois P. Heinz_, Jul 20 2019