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A213211
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Triangular array read by rows: T(n,k) is the number of size k subsets of {1,2,...,n} such that (when the elements are arranged in increasing order) the smallest element is congruent to 1 mod 3 and the difference of every pair of successive elements is also congruent to 1 mod 3.
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1
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 2, 3, 4, 1, 1, 1, 1, 3, 3, 4, 5, 1, 1, 1, 1, 3, 6, 4, 5, 6, 1, 1, 1, 1, 3, 6, 10, 5, 6, 7, 1, 1, 1, 1, 4, 6, 10, 15, 6, 7, 8, 1, 1, 1, 1, 4, 10, 10, 15, 21, 7, 8, 9, 1, 1, 1, 1, 4, 10, 20, 15, 21, 28, 8, 9, 10, 1, 1, 1
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OFFSET
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0,12
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COMMENTS
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REFERENCES
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Combinatorial Enumeration, I. Goulden and D. Jackson, John Wiley and Sons, 1983, page 56.
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LINKS
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FORMULA
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G.f.: (1 + x + x^2)/(1 - x^3 - y*x).
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EXAMPLE
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T(6,3) = 4 because we have: {1,2,3}, {1,2,6}, {1,5,6}, {4,5,6}.
1;
1, 1;
1, 1, 1;
1, 1, 1, 1;
1, 2, 1, 1, 1;
1, 2, 3, 1, 1, 1;
1, 2, 3, 4, 1, 1, 1;
1, 3, 3, 4, 5, 1, 1, 1;
1, 3, 6, 4, 5, 6, 1, 1, 1;
1, 3, 6, 10, 5, 6, 7, 1, 1, 1;
1, 4, 6, 10, 15, 6, 7, 8, 1, 1, 1;
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MAPLE
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T:= (n, k)-> binomial(k+floor((n-k)/3), k):
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MATHEMATICA
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nn=10; f[list_]:=Select[list, #>0&]; Map[f, CoefficientList[Series[ (1+x+x^2)/(1-x^3-y x), {x, 0, nn}], {x, y}]]//Grid
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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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