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A229706
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Triangular array read by rows: T(n,k) is the number of unimodal compositions of n with greatest part equal to k; n>=1, 1<=k<=n.
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2
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1, 1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 6, 5, 2, 1, 1, 9, 9, 5, 2, 1, 1, 12, 16, 10, 5, 2, 1, 1, 16, 25, 19, 10, 5, 2, 1, 1, 20, 39, 32, 20, 10, 5, 2, 1, 1, 25, 56, 54, 35, 20, 10, 5, 2, 1, 1, 30, 80, 84, 61, 36, 20, 10, 5, 2, 1, 1, 36, 109, 129, 99, 64, 36, 20, 10, 5, 2, 1
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OFFSET
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1,5
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COMMENTS
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A unimodal composition is a composition such that for some j, m, 1 <= x(1) <= x(2) <= ... <= x(j) >= x(j+1) >= ... >= x(m) >= 1.
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REFERENCES
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E. M. Wright, Stacks, Quart. J. Math. Oxford 19 (1968) 313-320, table s(r).
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LINKS
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FORMULA
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O.g.f. for column k: x^k/prod(i=1..k-1, 1-x^i )^2.
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EXAMPLE
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1;
1, 1;
1, 2, 1;
1, 4, 2, 1;
1, 6, 5, 2, 1;
1, 9, 9, 5, 2, 1;
1, 12, 16, 10, 5, 2, 1;
1, 16, 25, 19, 10, 5, 2, 1;
1, 20, 39, 32, 20, 10, 5, 2, 1;
1, 25, 56, 54, 35, 20, 10, 5, 2, 1;
T(5,3) = 5 because we have: 3+2 = 2+3 = 3+1+1 = 1+3+1 = 1+1+3.
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MAPLE
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b:= proc(n, t, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
`if`(k>0, `if`(n<k, 0, add(b(n-j, j, `if`(j=k, 0, k)),
j=t..min(k, n))), add(b(n-j, j, 0), j=1..min(t, n))))
end:
T:= (n, k)-> b(n, 1, k):
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MATHEMATICA
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Map[Select[#, #>0&]&, Drop[Transpose[Table[CoefficientList[Series[x^n/(1-x^n)/Product[1-x^i, {i, 1, n-1}]^2, {x, 0, nn}], x], {n, 1, nn}]], 1]]//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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