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A309280
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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.
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13
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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, 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, 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
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OFFSET
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1,2
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COMMENTS
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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.
The sequence of column k satisfies a linear recurrence with constant coefficients of order 3*A000593(k).
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LINKS
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FORMULA
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T(n+1,n*(n+1)/2+1) = A000009(n) for n >= 0.
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EXAMPLE
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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.
Triangle T(n,k) begins:
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, 2, 2, 1, 1, 1;
...
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MAPLE
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b:= proc(n, m, s) option remember; `if`(n=0, [`if`(s=0, 1, 0), 0],
b(n-1, m, s) +(g-> g+[0, g[1]*n])(b(n-1, m, irem(s+n, m))))
end:
T:= (n, k)-> b(n, k, 0)[2]/k:
seq(seq(T(n, k), k=1..n*(n+1)/2), n=1..10);
# second Maple program:
b:= proc(n, s) option remember; `if`(n=0, add(s/d *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);
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MATHEMATICA
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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[n_, k_] := b[n, k, 0][[2]]/k;
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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